A DIMENSION-FREE LOGARITHMIC SOBOLEV INEQUALITY
FOR SUBGAUSSIAN LOG-CONCAVE MEASURES
An expository rewrite: one theorem, every step, in plain language.
Notation used throughout (all ASCII, so it copies cleanly):
R^n n-dimensional Euclidean space (n can be any positive
integer; nothing below depends on how big n is).
|x| length of the vector x; x.y = dot product.
D gradient (the vector of first partial derivatives).
D^2 Hessian (the matrix of second partial derivatives).
Delta Laplacian (the trace of the Hessian).
int f dmu the integral of f against the measure mu.
mu(A) probability that a mu-random point lands in the set A.
A measure mu on R^n is a probability measure when mu(R^n)=1. It has a
density rho when mu(A) = int_A rho(x) dx for every set A; then rho is a
nonnegative function integrating to 1. "Full-dimensional support" means
the set where rho>0 is not flat (it has volume in every direction).
1. THE CLAIM, IN PLAIN WORDS
1.1 Entropy.
For a nonnegative function g and a probability measure mu, the entropy
of g with respect to mu is
Ent_mu(g) = int g log g dmu - (int g dmu) log( int g dmu ),
with the convention 0 log 0 = 0. Entropy measures how uneven g is: it
is zero exactly when g is constant, and it grows as g becomes spiky.
The subtraction term is there so that adding a constant to log g, or
rescaling g by a positive factor, changes nothing.
1.2 Dirichlet energy.
For a smooth function f, its Dirichlet energy with respect to mu is
int |Df|^2 dmu,
the average squared steepness of f. It is zero exactly when f is
constant. It measures how much f wiggles.
1.3 The logarithmic Sobolev inequality (LSI).
Saying mu satisfies an LSI with constant R means
Ent_mu(f^2) <= 2 R int |Df|^2 dmu for all smooth f,
again with constant functions giving equality at zero. In words: if f
wiggles only a little (small energy), then f^2 cannot be very uneven
(small entropy). The factor 2 is a convention; R is what carries the
information. The smallest such R is written R(mu).
1.4 Why "dimension-free" is the whole point.
Many inequalities in n dimensions carry constants that blow up like n,
sqrt(n), or worse as n grows. A constant that is completely independent
of n is called dimension-free, or absolute if it is a fixed number like
17. Bounds whose constants grow with n become useless for the
high-dimensional objects (convex bodies, machine-learning models,
statistical estimators) that motivate the question. We want one fixed
number that works in every dimension.
1.5 Log-concave measures.
A density rho on R^n is log-concave when rho = exp(-V) for a convex
function V, on the set where rho>0. Convexity of V means its graph
bends upward; equivalently the superlevel sets {rho >= t} are convex.
Examples include the uniform measure on any convex body (a ball, a
cube, a simplex), the Gaussian, the exponential, and products and
Gaussian convolutions of all of these. Log-concave measures are the
standard model for "a probability sitting on a convex region and
decaying outward without bumps."
1.6 Subgaussian linear functionals.
Fix a centered random vector X with law mu (centered means E X = 0).
For a unit vector theta, the scalar theta.X is one linear shadow of X.
We call X linearly a-subgaussian when
E exp( (theta.X)^2 / a^2 ) <= 2 for every unit vector theta.
This says every one-dimensional shadow has tails no heavier than a
Gaussian of variance about a^2: it controls all the linear functions of
X at once, but only the linear ones. The parameter a is exactly the
"width" of these shadows.
The question the paper answers: do these Gaussian tails of the LINEAR
shadows already force an LSI for ALL (including nonlinear) test
functions, with a constant independent of dimension?
2. THE THEOREM AND ITS TRANSPORT CONSEQUENCE
2.1 The main theorem.
Theorem 1 (Main). There is an absolute constant C < infinity with the
following property. For every n >= 1, every centered log-concave
probability measure mu on R^n that has a Lebesgue density, and every
a > 0 such that a random vector X with law mu is linearly a-subgaussian,
Ent_mu(f^2) <= C a^2 int |Df|^2 dmu for every smooth compactly
supported f on R^n.
The same C works in every dimension and for every such mu.
In the convention of Section 1.3 (Ent(f^2) <= 2 R(mu) int|Df|^2), the
theorem says R(mu) <= (C/2) a^2: the logarithmic Sobolev constant is
bounded by a universal multiple of the squared subgaussian width. Note
what is NOT assumed: the density need not be smooth, and no positive
curvature (no D^2 V >= kappa > 0) is required. So uniform measures on
convex bodies, whose potentials are flat and only convex, are included.
Before this result, only a dimension-dependent bound like R(mu) <= C n
a^2 or C sqrt(n) a^2 was known in general.
2.2 Why the converse is free (so the hypothesis is sharp).
The LSI already forces Gaussian tails for 1-Lipschitz functions by the
classical Herbst argument; in particular it forces exactly the
subgaussian bound on linear functions that Theorem 1 assumes. So the
hypothesis "linear functions are a-subgaussian" is, up to universal
constants, also necessary for an LSI with constant of order a^2. The
theorem is therefore not wasting any assumption: it upgrades a
necessary condition to a sufficient one, at the same scale.
2.3 A consequence for transport.
Two standard objects. For two probability measures alpha, beta on R^n
with finite second moment, the squared 2-Wasserstein distance is
W2(alpha,beta)^2 = inf over couplings pi of int |x-y|^2 dpi(x,y),
where a coupling is a joint law with the two given marginals. It is the
cheapest way to move the mass of alpha onto beta, paying squared
distance. The relative entropy of alpha with respect to beta is
H(alpha | beta) = int p log p dbeta when alpha = p beta,
= +infinity when alpha is not absolutely
continuous with respect to beta.
Corollary 1 (Quadratic transport-entropy inequality). Under the
hypotheses of Theorem 1, every probability nu with finite second moment
satisfies
W2(nu,mu)^2 <= C a^2 H(nu | mu),
with the same absolute constant C. This follows from the LSI through the
Otto-Villani theorem, which converts any LSI into a
transport-entropy inequality with matching constants; the proof at the
level of this document is in Section 5.7 and only needs an
approximation argument to remove the smoothness of the density.
The corollary is the useful form in applications: it says it costs very
little mass-movement energy (order a^2 times a bit of relative entropy)
to reach mu from any competitor nu. This is what turns "control of
linear functions" into concrete concentration statements.
3. HISTORY AND WHY IT MATTERS
3.1 The inequality and the Gaussian.
The logarithmic Sobolev inequality was introduced by Leonard Gross in
1975 in his study of hypercontractivity, the property that a semigroup
pushes functions into higher integrability. For the standard Gaussian
measure the LSI holds with R = 1, independent of dimension, and Gross
showed the constant cannot be improved. This paper asks how far the
"constant 1, no dimension" phenomenon survives beyond the Gaussian.
3.2 The curvature approach and its blind spot.
Bakry and Emery (1985) gave a powerful criterion. Write the density as
rho = exp(-V). Suppose the potential is strictly convex in the strong
sense D^2 V >= kappa Id with kappa > 0. Then mu satisfies the LSI with
R <= 1/kappa. The proof runs a diffusion whose drift is D V and shows
the flow contracts distances at rate kappa.
The catch: log-concavity says only D^2 V >= 0, i.e. kappa = 0. The
curvature criterion then gives nothing finite. A uniform measure on a
convex body is the extreme case: its potential is infinite outside the
body and flat inside, so it has zero curvature in the interior and
infinite curvature on the boundary. The Bakry-Emery machine is blind to
exactly the examples we care about. The whole difficulty is to replace a
local curvature hypothesis by the global tail hypothesis that linear
functions are subgaussian.
3.3 Necessity and the earlier dimension-dependent bounds.
The Herbst argument (see Ledoux's 1999 survey) shows that an LSI with
constant R gives Gaussian concentration: for every 1-Lipschitz u,
log E exp( t (u - E u) ) <= R t^2 / 2.
Applying this to linear functions, an LSI with constant of order a^2
forces the linear subgaussian bound with width of order a. So some
subgaussian control of linear functions is necessary. The open problem
was sufficiency: does it give the LSI back, dimension-free?
Sergey Bobkov (1999) gave a norm-based criterion: the entropy constant
is bounded by a universal multiple of the squared subgaussian norm of
the scalar |X|. Combining this with the linear subgaussian assumption,
which controls |X| only through a Holder step E exp(|X|^2/(n a^2)) <= 2,
yields an entropy constant of order n a^2, with a dimension factor.
Pierre Bizeul (conjectured in his 2023 preprint, cited here as the 2026
journal version) formulated precisely that linear subgaussian tails
should suffice for a dimension-free LSI within the log-concave class.
Bizeul proved an improvement with constant of order sqrt(n) a^2 in
general, and the fully dimension-free statement in the special case of
rotationally invariant log-concave measures. The survey of Klartag and
Lehec (2024/2025) restates the conjecture together with these bounds.
3.4 What is genuinely new.
The linear subgaussian hypothesis controls only one-dimensional shadows,
while the LSI conclusion controls the entropy of genuinely nonlinear
test functions. The gap between "linear control" and "nonlinear
conclusion" is where the entire argument lives. Emanuel Milman (2010)
had shown that for log-concave measures, Gaussian concentration for all
Lipschitz functions is equivalent to Gaussian isoperimetry, which in
turn gives an LSI; but "all Lipschitz functions" is much stronger than
"linear functions." The theorem closes the remaining gap: linear
functions turn out to already force the concentration of all Lipschitz
functions, hence the LSI. Theorem 1 is the first proof that the
conjecture is true in full generality.
4. THE PLAN OF THE PROOF (READ THIS FIRST)
The proof is a proof by contradiction. We assume Theorem 1 fails and
derive nonsense. Everything technical is in service of one idea: a
counterexample that violates the LSI would have to behave like a
Gaussian in a very rigid way, both from above (entropy contraction) and
from below (an information test), and those two descriptions cannot
both hold.
4.1 Assume a counterexample, then clean it up.
Failure of Theorem 1 gives a sequence of centered, compactly supported,
full-dimensional log-concave laws mu such that the LSI constant R(mu)
is normalized to 1 while the linear subgaussian width a tends to 0. The
ratio R(mu)/a^2 then blows up; that is the failure. Two harmless
operations keep the failure: restricting to a large ball and tilting by
exp(-epsilon |x|^2 / 2). They keep the law log-concave, compactly
supported, and smooth enough to work with.
4.2 Gaussian regularization.
Convolve mu with a tiny Gaussian of variance s. Write
mu_s = law(X + sqrt(s) G), with G standard Gaussian independent of X,
and fix s = 1/8 once and for all. Convolution makes the density smooth
and strictly positive and keeps it log-concave (a Gaussian is
log-concave and a convolution of log-concave densities is
log-concave). Its subgaussian width a_s still tends to 0. Crucially,
the regularized LSI constants R_s tend to 1, because s is fixed and
small. So from now on we have a smooth, positive, log-concave law mu_s
sitting at the edge: it is one unit of entropy-constant away from a
counterexample.
4.3 The critical law and a near-first-eigenfunction.
From mu_s we build a second probability eta and a vector field w. There
are two cases, and we must handle both because we cannot predict which
occurs:
Strict branch. The entropy quotient is attained at a positive
optimizer f solves the logarithmic Euler equation. Put
eta = f^2 mu_s and w = D log f^2. Then eta satisfies its own LSI
with constant <= 1/lambda, and w is an approximate first
eigenfunction of the diffusion A_eta = -Delta - D log(eta).D:
(A_eta - 1) w is small in L^2, E_eta|w|^2 = I > 0.
Gap branch. The LSI and Poincare constants coincide. Then w = D phi,
where phi is an eigenfunction of A_{mu_s} with eigenvalue lambda, and
eta = mu_s. The same near-eigenfunction relation holds.
In both cases eta satisfies a logarithmic Sobolev inequality with a
constant close to 1, and w is close to a first eigenfunction of its
diffusion. That is the single object the rest of the proof works with.
4.4 The prediction curve.
Add fresh Gaussian noise. For r > 0 form U_r = Y + sqrt(r) G with
Y ~ eta and G standard, and let m(r) be the fraction of w that the best
predictor can recover from U_r:
m(r) = (1/I) int |E[w(Y) | U_r]|^2 deta.
If w were an exact first eigenfunction, the semigroup would give
m(r) = e^{-r} exactly. We prove this is essentially true anyway:
Theorem (prediction curve). For every fixed 0 < r < 1/4,
m(r) -> e^{-r} along the counterexample sequence.
The proof pairs two facts. Upper bound: entropy contraction of the
diffusion (the LSI of eta) caps how much information about w can
survive the noise. Lower bound: an entropy-loss comparison lemma
tracks the loss under repeated Gaussian observations; observing many
independent copies lets us throw away at most one direction per copy,
accumulating negligible cost. Together they pin m(r) to e^{-r}.
4.5 From the curve to tensors.
The exact identity m(r) = e^{-r}, integrated against Gaussian Hermite
polynomials, says the prediction is Bessel-tight at every derivative
order. This is the bridge from a statement about the vector w to a
statement about all its derivatives. It produces symmetric tensor
fields T_l of order l+1 with
||T_l||^2 = (1+o(1)) I, ||DT_l - T_{l+1}|| = o(sqrt I),
||(A-1) T_l|| = o(sqrt I), <T_l, K T_l> = o(I),
where K = D^2(-log mu_s) is the curvature. So derivatives keep the
same size, stay nearly eigenfunctions, and avoid the curved directions
of mu_s. This holds simultaneously for any fixed finite list of levels.
4.6 Matrices and the frozen law.
Pack the tensors of a band of levels into a big column Psi, separate
one slot, and let
rho = Psi Psi^*, R = rho^{1/2}
be a positive matrix field. Many symmetric slots let us average the
action over slots with only O(1/k) error, and shifting the band costs
only its endpoints; this yields sharp energy: ||DR||^2 -> 1. From the
derivative-action estimate one builds a bounded symmetric matrix column
H_i with a Sylvester-type relation D_i R ~ (H_i R + R H_i)/2 and almost
commuting H_i. Then, running the stationary diffusion for time t and
freezing H_i at the initial point, R(Y_t) is described by a matrix
lognormal:
sigma = direct-sum over spectral blocks beta^2 exp( sqrt(8t) H[N] - 4t ),
with N standard Gaussian. This is the frozen diffusion law. It is the
engine of the final contradiction, and it is where the many symmetric
tensor slots earn their keep: they make H[N] behave like a scalar
Gaussian even though the H_i need not commute.
4.7 The information contradiction.
We now observe a Gaussian vector whose covariance is a bounded
increasing function of R(Y_t)^2, with a scale T:
Z_T = ( Id + f_L(T R(Y_t)^2) )^{1/2} G, f_L(q) = q/(1+q/L).
Upper bound (from the LSI): the mutual information between the state
and the observation, integrated against dT/T^2, is at most a universal
constant. The factor dT/T^2 is the load-bearing trick. In a spectral
block of size beta^2 the substitution x = T beta^2 gives
dT/T^2 = beta^2 dx/x^2,
assigning to each block exactly the weight beta^2 that the matrix
(frozen) trace uses, even when the spectrum is arbitrarily diffuse.
Lower bound (from the frozen law): on the same event, the observation
reproduces a scalar Gaussian variance-mixture experiment. The entropy
of that experiment, integrated over the same scales, grows without bound
once the diffusion time t and the covariance cap L are taken large
enough.
Upper bound is a constant, lower bound is unbounded. Contradiction.
4.8 The order of limits.
Throughout, every auxiliary parameter (time t, cap L, truncation m,
band size k) is fixed BEFORE the counterexample index is sent to
infinity. Only after that sequence limit do we let t and L grow. No
uniformity in the growing parameters across the sequence is ever
required. This order is repeated where it matters so it is never in
doubt.
4.9 Reading order for the technical part.
Section 5 analytic toolkit: entropy, information, curvature,
diffusion, entropy dissipation, Gaussian channels.
Section 6 normalization and construction of the critical law
(both branches).
Section 7 the prediction curve.
Section 8 the tensor hierarchy.
Section 9 the matrix square root and the frozen law.
Section 10 the covariance observations and the final information
contradiction, then the assembly of Theorem 1.
Figure 1 (description): two arrows from "strict branch" and "gap
branch" merge into "common critical law and field." From there the
prediction curve feeds the tensor hierarchy, the hierarchy feeds the
matrix frozen law, and the frozen law plus the LSI both feed the
information contradiction. Only Theorem 1 needs the two branches to
rejoin at the end.
5. TOOLKIT: ENTROPY, DIFFUSION, AND GAUSSIAN CHANNELS
This section sets up the analytic language used in every proof that
follows. Nothing here is specific to the counterexample; it is the common
vocabulary.
5.1 Two time conventions that set all factors.
We will run two different "time" evolutions and must not confuse them.
(a) Gaussian convolution. Convolving a law with a centered Gaussian of
covariance r Id has generator (1/2) Delta. Its time parameter is r,
a variance.
(b) Stationary diffusion. A reversible diffusion with invariant law pi
and generator -A_pi, with noise coefficient sqrt(2). Its time
parameter is a genuine clock time.
The two conventions differ by factors of 2 throughout. We fix them once
and state every dissipation identity in the convention used, so no
factor can drift.
5.2 Relative entropy, Fisher information, and the two constants.
For probabilities p, pi with p absolutely continuous relative to pi,
H(p | pi) = int log(dp/dpi) dp, H(p | pi) = +infinity otherwise.
The Fisher information of p relative to pi is
Ical(p | pi) = 4 int |D sqrt(dp/dpi)|^2 dpi.
When the density is positive and smooth this equals
int |D log(dp/dpi)|^2 dp: the average squared score. The value is
+infinity unless the square root of the density lies in the weighted
Sobolev space.
The logarithmic Sobolev constant R(pi) and the Poincare constant
R^P(pi) are defined by the best constants in
Ent_pi(f^2) <= 2 R(pi) int |Df|^2 dpi,
Var_pi(f) <= R^P(pi) int |Df|^2 dpi, Var_pi(f)=int f^2 dpi - (int f dpi)^2.
Equivalently, in "relative" form, the LSI constant R = R(pi) is the
smallest number with
2 H(p | pi) <= R Ical(p | pi) for all p.
The Dirichlet form is int |Df|^2 dpi. Constant functions have zero
entropy and zero energy, which is why the entropy is measured relative
to the mean.
Linearization: expanding Ent about f = 1 + epsilon g to second order in
epsilon gives the variance bound, so R^P(pi) <= R(pi). The Poincare
constant is never larger than the log-Sobolev constant, as one expects:
a variance bound is weaker than an entropy bound.
5.3 Sobolev space and why approximations are allowed.
For a smooth positive density the weighted Sobolev space is the
completion of the smooth compactly supported functions in the norm with
squared norm int (f^2 + |Df|^2) dpi. Cutoffs, truncation of values, and
smoothing on compact sets show this is also the space of functions whose
weak gradient is square-integrable. A compact-test LSI extends to the
whole space by lower semicontinuity of the entropy and the Fisher
information, and by approximation. Every inequality below stated for
smooth compact tests therefore holds on this space; we use this silently.
5.4 The entropy-duality formula.
For a probability pi, one exact identity underpins most of the analysis:
H(p | pi) = sup over bounded g of { int g dp - log int e^g dpi }.
Jensen's inequality gives the upper bound (log int e^g dpi is an
exponential-moment), and bounded truncations of log(dp/dpi) give
equality. Three consequences we will use.
(i) Data processing. Pushing a probability through a measurable map,
or averaging through a Markov kernel, never increases relative
entropy. This is conditional Jensen.
(ii) Chain rule. Disintegrating a joint density and splitting its
logarithm gives
H(P_{U,Z} | Q_U x nu) = H(P_U | Q_U) + E_{P_U} H(P_{Z|U} | nu),
with conditional entropies averaged over U. Mutual information is
I(U;Z) = H(P_{U,Z} | P_U x P_Z).
(iii) Mean lower bound. Taking g(z)=v.z in the duality formula and
optimizing over v gives H(p | gamma) >= |E_p Z|^2/2 for a Gaussian
gamma of unit covariance: relative to a Gaussian, an entropy
budget bounds how far the mean can be pushed.
5.5 Positive curvature: the Bakry-Emery lemma.
Lemma 1 (Positive curvature). Let S be a positive definite matrix on R^n,
and let U be a proper lower semicontinuous convex function, allowed to be
+infinity (so the support may be a convex set with boundary). Suppose
pi(x) = Z^{-1} exp( -(1/2) x.Sx - U(x) ) dx
is a probability with full-dimensional support. Then
Ent_pi(f^2) <= 2 E_pi < Df, S^{-1} Df >,
Var_pi(f) <= E_pi < Df, S^{-1} Df >,
first for compact smooth f, then for all f in the form domain.
Proof. A linear change of variables reduces to S = Id. First suppose
V(x) = |x|^2/2 + U(x) is smooth with bounded Hessian. Run the diffusion
with generator L = Delta - DV.D, drifting with -DV; it is reversible
with invariant law pi because the weight exp(-V) makes DV the score.
Let J_t be the derivative of the flow with respect to the starting
point. Differentiating the flow equation gives J'_t = -D^2 V(X_t) J_t,
so with D^2 V >= Id we get ||J_t|| <= e^{-t}: the flow contracts
distances exponentially. Weighted Cauchy-Schwarz then gives, for a
smooth positive q bounded away from zero,
|D P_t q|^2 / (P_t q) <= e^{-2t} P_t ( |Dq|^2 / q ),
where P_t = e^{tL} is the semigroup. Integration by parts yields
d/dt int (P_t q) log(P_t q) dpi = - int |D P_t q|^2 / (P_t q) dpi.
Couple a trajectory started at x to a stationary trajectory driven by
the same Brownian motion; the contraction bound says the distance
shrinks like e^{-t}. Since pi has Gaussian tails, P_t q -> E_pi q and
its entropy -> 0. Integrating the two displayed identities from 0 to
infinity gives Ent_pi(q) <= (1/2) int |Dq|^2/q dpi. Put q = f^2+eps and
send eps -> 0.
For general U, replace U by its quadratic infimal convolutions
U_j(x) = inf_z { U(z) + j|x-z|^2 }. These are convex, increase to U,
and have Lipschitz gradients, so U_j is smooth enough after mollifying
at radii -> 0. The affine minorant of U gives a uniform Gaussian-times-
linear domination, the LSI passes to the limit, and the boundary of a
full-dimensional convex support has zero Lebesgue measure so boundary
values do not matter. Inserting f = 1 + eps g and comparing eps^2
coefficients yields the variance inequality. QED.
The lemma is the entire content of the curvature method: curvature
D^2 V >= Id buys an LSI with constant 1. A "minimum curvature kappa"
version rescales to constant 1/kappa. And it is exactly this hypothesis
that log-concavity (D^2 V >= 0) fails to provide.
5.6 Products and Gaussian convolution add the constants.
Suppose independent pieces pi_1, pi_2 satisfy LSIs with energy forms
Q_1, Q_2. Their product satisfies the LSI with energy Q_1 + Q_2. Indeed
decompose the entropy of f on the product into conditional entropy on
the fiber plus entropy of the fiber integral, and use
|D_x ( int f(x,y)^2 dpi_2(y) )^{1/2}|^2 <= int |D_x f(x,y)|^2 dpi_2(y).
The variance version uses conditional variance and differentiation of
the fiber mean. In particular, for independent X and Z,
R( law(X+Z) ) <= R( law(X) ) + R( law(Z) ).
The Gaussian of variance s Id has R = s; adding Gaussian noise adds s to
the constant. This is why the regularized constants R_s in Section 4.2
obey R_s <= 1 + s.
5.7 Herbst's argument: entropy control forces concentration.
Lemma 2 (Herbst). If pi satisfies the LSI with constant R, then every
integrable 1-Lipschitz function u satisfies
log E_pi exp( t (u - E_pi u) ) <= R t^2 / 2 for all real t.
Proof. Apply the LSI to e^{tu/2}, which has D(e^{tu/2}) = (t/2) e^{tu/2}
Du. Writing psi(t) = log E_pi e^{tu}, the LSI reads
t psi'(t) - psi(t) <= R t^2 / 2. That is the derivative of psi(t)/t:
integrate from 0 to t using psi(0)=0 and continuity. Bounded Lipschitz
truncations and Fatou extend to all integrable 1-Lipschitz u. QED.
Distance to the mean under pi therefore has a subgaussian tail of
variance R. Conversely, if every 1-Lipschitz function has a subgaussian
tail of variance R, this is "concentration," and it is one route back to
the LSI (Milman's theorem, cited in Section 3). Applied to linear
functions u(x) = theta.x, Herbst says an LSI of constant R forces the
linear subgaussian bound of Section 1.6 with a of order sqrt(R). This is
the sharpness statement of Section 2.2.
5.8 The Dirichlet operator and the stationary diffusion.
Lemma 3 (Dirichlet operator). Let pi be a smooth strictly positive
probability density on R^n. On smooth compactly supported functions
define
A_pi = - Delta - D log(pi) . D.
This nonnegative symmetric operator is essentially self-adjoint in
L^2(pi), and smooth compact functions are a core for its closure. In
particular, if f in L^2(pi) has A_pi f (in the distributional sense)
in L^2(pi), then f is in the domain of the closed operator. The
semigroup P_u = e^{-u A_pi} preserves constants and is realized by the
stationary diffusion
dY_u = sqrt(2) dB_u + D log pi(Y_u) du, Y_0 ~ pi.
For f in the domain, the stationary Ito formula holds in L^2, with drift
- A_pi f and martingale integrand sqrt(2) Df.
Proof sketch, complete in its logical steps. Integration by parts gives
<f, A_pi g> = int Df . Dg dpi, so A_pi is symmetric and nonnegative. If
v in L^2(pi) is orthogonal to (1 + A_pi) of all smooth compact
functions, then (1 + A_pi) v = 0 distributionally. Conjugating by
sqrt(pi) reduces this to a Poisson equation with a smooth local
potential, and local elliptic bootstrap makes v smooth. Testing against
chi_R^2 v with a cutoff chi_R equal to 1 on a ball of radius R and
|D chi_R| <= C/R, and using 2ab <= a^2/2 + 2b^2, gives
int chi_R^2 v^2 dpi + (1/2) int chi_R^2 |Dv|^2 dpi <= 2 int |D chi_R|^2 v^2 dpi,
which forces v = 0. Hence the range of 1 + A_pi is dense. The bound
||(1+A_pi)f||_2 >= ||f||_2 says the closure's range is also closed, so
it is all of L^2; the closure is self-adjoint (solve the adjoint
equation and use the orthogonal complement argument). This proves the
domain and self-adjointness claims. The cutoffs converge to 1 while
their energy tends to 0, so 1 is in the kernel and P_u 1 = 1. Solving
the stochastic equation with killing on balls, whose kernels are
symmetric sub-Markov by the Dirichlet heat equation, and increasing the
balls, identifies the minimal semigroup with P_u; essential
self-adjointness shows no mass is lost, so the diffusion is stationary
and non-exploding. QED.
The point of the lemma: it gives a genuine reversible Markov process
with known generator and a well-defined energy form, on which "time"
entropy gradients are computable. Everything downstream is a statement
about this process.
5.9 Entropy dissipation (de Bruijn identities).
Lemma 4 (Entropy dissipation). For the semigroup of Lemma 3, if
p_0 = q_0 pi has finite relative entropy and p_u = (P_u q_0) pi, then
for 0 <= a <= b,
H(p_a | pi) - H(p_b | pi) >= int_a^b Ical(p_u | pi) du.
If instead two probabilities both evolve by Gaussian convolution,
p_r = p_0 * gamma_r and q_r = q_0 * gamma_r, then
H(p_a | q_a) - H(p_b | q_b) >= (1/2) int_a^b Ical(p_r | q_r) dr
whenever the left-hand entropy at time a is finite; here gamma_r is the
centered Gaussian of covariance r Id, and a = 0 is allowed. For the
standard Gaussian Ornstein-Uhlenbeck semigroup the first inequality is
an equality, including at time 0 when the initial entropy is finite.
All assertions hold after averaging over conditioning labels whose
noises are independent given those labels.
Proof. For a bounded density q bounded away from zero, the form chain
rule, first for smooth functions and then by form approximation, gives
(d/du) int (P_u q) log(P_u q) dpi = - int |D P_u q|^2 / (P_u q) dpi.
Spectral smoothing puts P_u q in the operator domain for u > 0.
Approximating an arbitrary finite-entropy density from above and below
and normalizing, the initial entropies converge; L^1 contraction keeps
later densities close; lower semicontinuity of entropy and of the
Fisher information, plus Fatou in time, gives the first display from
time 0. Restarting at a gives arbitrary a <= b.
For simultaneous heat evolution put z = p_r / q_r. Direct
differentiation with (d/dr) p_r = Delta p_r / 2 gives, for convex
Phi_k increasing to z log z - z + 1,
(d/dr - (1/2) Delta) { q_r Phi_k(z) } = -(1/2) q_r Phi_k''(z) |Dz|^2.
Integrate against spatial cutoffs, let the cutoff go to infinity, then
k -> infinity, and use monotone convergence. Data processing gives
H(p_eps | q_eps) <= H(p_0|q_0), so the initial time is included. For
Ornstein-Uhlenbeck evolution relative to a standard Gaussian,
generator Delta - x.D, the same convex truncation stacked with expanding
radial cutoffs gives equality on positive time; relative entropy is
continuous at time 0 by data processing plus lower semicontinuity.
Tonelli handles conditional versions. QED.
In words: relative entropy is nonincreasing under both evolutions, and
its rate of decrease equals the Fisher information. This is the "engine"
that converts an entropy budget into an integral information budget.
5.10 The Gaussian channel identity (Guo-Shamai-Verdu).
This is the standard "I-MMSE" identity: the derivative of entropy
through a Gaussian channel is half the power of the minimum-mean-square
predictor. We need its conditional, time-integrated form in our
conventions.
Lemma 5 (Gaussian signals). Let S be an R^d-valued random vector with
finite second moment, let U be any side information, and let G be a
standard Gaussian independent of (S,U). Write Z_t = sqrt(t) S + G and
d(t) = E_U H( Law(Z_t | U) | gamma ),
the conditional relative entropy against the standard Gaussian gamma,
averaged over the side information. Then d(0) = 0, d(t) <= t E|S|^2/2,
and
d(t) = (1/2) int_0^t E |E[S | Z_v, U]|^2 dv.
The integrand is nondecreasing in v. Taking U constant gives the
unconditional identity.
Proof. Conditional convexity of relative entropy and the shift formula
H( gamma(. - sqrt(t) s) | gamma ) = t|s|^2/2 give the upper bound.
Gaussian kernel differentiation gives the score of the output,
D_z log( dLaw(Z_t|U) / dgamma )(z) = sqrt(t) E[S | Z_t = z, U].
This is justified at every t > 0 by the Gaussian kernel and the finite
second moment, or by truncating S. Running Ornstein-Uhlenbeck evolution
for time u replaces t by t e^{-2u}. Its entropy dissipation identity
together with dt/du = -2t gives d'(t) = (1/2) E |E[S|Z_t,U]|^2 almost
everywhere on (0,infinity). The upper bound gives d(t) -> 0 at zero,
so integrating from 0 proves the identity. Monotonicity of the
prediction energy: the channel at v < t is obtained from the channel at
t by multiplying the output by sqrt(v/t) and adding fresh Gaussian noise
of covariance (1 - v/t) Id, and conditional Jensen gives
nondecreasingness. QED.
The lemma turns a Gaussian observation into a computing device: the
information it carries is exactly the integral of the squared best
prediction of the signal. We will use it both directions - to bound
information from a predictor, and to read off a predictor from
information.
5.11 The transport consequence of the LSI.
Proof of Corollary 1. Jensen in each coordinate gives
int |x|^2 dmu <= n a^2 log 2, so mu has finite second moment. Only the
case H(nu|mu) < infinity needs proof. For s > 0 put gamma_s =
N(0, s Id), mu_s = mu * gamma_s, and nu_s = nu * gamma_s. The density of
mu_s is smooth and strictly positive.
Claim: mu_s is log-concave with convex potential V_s. For X ~ mu and an
independent standard Gaussian G, the posterior potential of X given
X + sqrt(s) G = y is the original convex potential plus
|x - y|^2/(2s), so Lemma 1 bounds the posterior covariance by s Id.
Hence the Hessian of V_s satisfies
D^2 V_s(y) = s^{-1} Id - s^{-2} Cov( X | X + sqrt(s) G = y ) >= 0,
and Gaussian differentiation is valid without compact support. So mu_s
is centered and log-concave, smooth.
Next, mu_s is a_s-subgaussian with a_s = a + beta_s and
beta_s = sqrt(8s/3). For a scalar standard Gaussian Z,
E exp( z^2 / beta_s^2 ) = 2. The elementary inequality
(u+t)^2/(a+beta_s)^2 <= (a/(a+beta_s)) u^2/a^2 + (beta_s/(a+beta_s)) t^2/beta_s^2
and Holder show that if X is a-subgaussian and sqrt(s)G has width
beta_s, then X + sqrt(s)G is a_s-subgaussian. Theorem 1 with the Sobolev
extension gives the LSI with constant C a_s^2.
In the conventions of Otto-Villani, Ent(f^2) <= L int|Df|^2 is LSI(2/L),
whose transport conclusion is W2^2 <= L H. Their theorem applies to the
smooth convex potential of mu_s, hence
W2(nu_s, mu_s)^2 <= C a_s^2 H(nu_s | mu_s) <= C a_s^2 H(nu | mu).
The last step is data processing for the common Gaussian kernel, as in
Section 5.4. Coupling a vector with itself plus independent Gaussian
noise gives W2(mu_s, mu) <= sqrt(n s) and the same for nu. Let
s -> 0 at fixed n, with a_s -> a. QED.
The whole proof of the corollary is therefore three moves: regularize
by Gaussian convolution (which buys smoothness and curvature), invoke
the main inequality plus Otto-Villani on the regularized law, and let
the regularization vanish. The only inputs are Lemmas 1 and the data
processing bound.
6. THE CRITICAL LAW AND A NEAR-FIRST-EIGENFUNCTION
Suppose Theorem 1 is false. This section converts "a counterexample
exists" into a single smooth object: a probability eta satisfying an LSI
with constant close to 1, plus a vector field w that is almost a first
eigenfunction of eta's diffusion. There are two constructions of (eta,w);
we keep both alive because we cannot predict which occurs. Every limit
in this section is along the counterexample sequence; we suppress its
index.
6.1 Normalized counterexamples.
Lemma 6 (Normalized counterexamples). Failure of Theorem 1 produces a
sequence of centered, compactly supported, full-dimensional log-concave
probabilities mu such that
R(mu) = 1, sup_{|theta|=1} E_mu exp( (X.theta)^2 / a^2 ) <= 2, a -> 0.
For an absolute constant C they also satisfy
log E_mu exp( v.X ) <= C a^2 |v|^2, Cov_mu(X) <= C a^2 Id.
Proof. Start from laws and compact smooth tests for which the ratio
(entropy)/(energy)/a^2 tends to infinity. Choose the tests with positive
energy: a smooth test of zero energy is constant on the connected
interior of the convex support, and its entropy then vanishes. Restrict
the law to a large ball; this preserves each test quotient arbitrarily
well. Multiply the restricted density by exp(-eps |x|^2/2) and
normalize, with eps > 0 small enough to preserve the quotient again.
Both operations keep log-concavity, the combined density relative to the
original law is bounded by 2, and Lemma 1 gives a finite LSI constant
for the resulting compact law.
A bounded density increase changes the exponential-square expectation by
at most that factor. If E exp(Z^2/A^2) <= 4, Jensen gives
E exp(Z^2/(2A^2)) <= 2. Centering costs only another absolute factor:
E Z^2 <= A^2 log 4 and (Z - E Z)^2 <= 2Z^2 + 2(E Z)^2. Apply this to
each unit linear form with the same constants. Translation preserves
the test quotient. Thus the centered compact laws still have finite
R with R/a^2 -> infinity. Dilation by R^{-1/2} makes R(mu) = 1.
For the moment bound: if a centered scalar Z obeys
E exp(Z^2/a^2) <= 2, then E|Z|^k <= 2 a^k Gamma(k/2+1) by integrating
its tail. Expansion of exp(tZ), with the linear term absent, gives
log E exp(tZ) <= C a^2 t^2 for a|t| <= c; for a|t| >= c use
tZ <= Z^2/a^2 + a^2 t^2/4. Apply to Z = X.theta. Differentiating the
centered moment generating function at zero gives the covariance bound.
QED.
6.2 Gaussian regularization and posterior identities.
Fix s = 1/8. Let G be a fresh standard Gaussian and set
mu_s = law( X + sqrt(s) G ), X ~ mu,
R_s = R(mu_s),
b_s(y) = E[ X | X + sqrt(s) G = y ] (posterior mean),
V_s(y) = - log mu_s(y), (potential of mu_s).
Gaussian differentiation and the posterior covariance give the three
basic identities
D V_s(y) = ( y - b_s(y) ) / s,
M_s(y) = D b_s(y) = Cov( X | X + sqrt(s) G = y ) / s,
K_s(y) = D^2 V_s(y) = ( Id - M_s(y) ) / s.
The posterior of X given y is the original convex potential plus
|x - y|^2/(2s); Lemma 1 (applied to its linear forms) bounds its
covariance by s Id. Hence
0 <= M_s <= Id, 0 <= K_s <= s^{-1} Id.
All these functions are smooth. Compact support of X also gives
|b_s| <= sup over the support of mu of |x|, so b_s is bounded.
The matrix field K_s is the curvature of mu_s. Much of the proof is
about showing that the near-eigenfield w avoids K_s; K_s is only
nonnegative and can be huge near the boundary of the support, which is
exactly why a uniform radius eigenvalue argument is unavailable and the
whole argument must be structured around "w sees almost no curvature."
6.3 The regularized constants barely change.
Lemma 7 (Bounds for regularized constants). The function s -> R_s is
nondecreasing and 1-Lipschitz on (0, infinity), and
1 <= R_s <= 1 + s.
Proof. The heat equation for mu_s is a continuity equation with
velocity D V_s / 2. On a compact interval of positive times this
velocity is globally Lipschitz and grows at most linearly, so its smooth
flow is a diffeomorphism carrying mu_s to mu_t. Its Jacobian J satisfies
J' = K_t J / 2. Since K_t >= 0 the flow expands distances, so its
inverse is 1-Lipschitz for t >= s. Transport a test function by that
inverse: its entropy is unchanged and its Dirichlet energy cannot
increase. Hence R_t >= R_s: monotone.
The convolution bound R_{s+h} <= R_s + h from Section 5.6 and
R_s <= R_0 + s = 1 + s give the upper bounds. Finally, for every compact
smooth test the entropy and the energy converge to those under mu as
s -> 0, so taking the supremum of the quotient gives
liminf_{s->0} R_s >= R(mu) = 1; monotonicity then forces the whole
range to start at >= 1. QED.
So for the fixed tiny regularization s = 1/8 we have R_s very close to
1, while each mu_s is smooth, positive, and log-concave with a
well-behaved diffusion. That is our working law.
6.4 Compactness and the Gaussian entropy defect.
To extract an optimizer we need two facts about the regularized law
nu = mu * gamma_s: its form domain is compactly embedded in L^2(nu),
and the LSI holds with a small additive defect that lets maximizing
sequences be compactified.
Lemma 8 (Compactness and Gaussian entropy defect). Fix s > 0 and a
compactly supported probability mu on R^n, and put nu = mu * gamma_s.
Its form domain embeds compactly into L^2(nu). For every eps > 0 there
is a finite C_eps, depending on this fixed law, such that
Ent_nu(u^2) <= 2(s+eps) ||Du||_2^2 + C_eps ||u||_2^2.
If u_j -> u in L^2(nu), ||u_j||_2 = 1, and ||D u_j||_2^2 -> E_* <
infinity, then
limsup_j Ent_nu(u_j^2) <= Ent_nu(u^2) + 2s ( E_* - ||Du||_2^2 ).
The sharp coefficient 2s governs escape into the Gaussian tails; the
additive constant governs maximizing sequences that would collapse to a
constant.
Proof. Write r = dnu/dgamma_s. The Gaussian kernel formula gives
D log r = b_s / s, which is bounded, and |log r(y)| <= C(1+|y|). For
v = u sqrt(r),
Ent_nu(u^2) = Ent_{gamma_s}(v^2) - int v^2 log r dgamma_s,
Dv = sqrt(r) ( Du + (1/2) u D log r ).
Gaussian integration by parts followed by Cauchy-Schwarz gives
int |y|^2 v^2 dgamma_s = s n ||v||_2^2 + 2s int v (y.Dv) dgamma_s
<= 2 s n ||v||_2^2 + 4 s^2 ||Dv||_2^2.
Hence the linear-growth entropy correction is bounded by
delta ||Dv||_2^2 + C_delta ||v||_2^2 for every delta > 0. Gaussian LSI
and the derivative formula, with another small Cauchy-Schwarz
parameter, give the defective inequality.
The same moment estimate shows tightness of the L^2(nu) mass of any
bounded form sequence. On a fixed ball the density is bounded above and
away from zero, so the ordinary local Rellich theorem gives L^2
precompactness; a diagonal extraction plus tightness give global
compactness.
For the sharp coefficient 2s in the entropy-defect display, set
v_{j,L} = (|u_j| - L)_+ and m_{j,L} = ||v_{j,L}||_2^2. Strong L^2
convergence gives lim_{L} sup_j m_{j,L} = 0. The energies of v_{j,L}
and |u_j| wedge L sum to ||D u_j||_2^2, so lower semicontinuity for the
clipped functions gives
lim_{L} limsup_j ||D v_{j,L}||_2^2 <= E_* - ||Du||_2^2.
For a nonnegative v with squared norm m <= 1, the negative part of
v^2 log v^2 has integral at most C sqrt(m) (use -z log z <= C sqrt(z) on
0 <= z <= 1), and m log m <= 0, so the defective inequality gives
int v^2 log_+ v^2 dnu <= 2(s+eps)||Dv||_2^2 + C_eps m + C sqrt(m).
If |u_j| > M L with M > 1 fixed then |u_j| <= c_M v_{j,L} with
c_M = M/(M-1). Passing bounded truncations of z^2 log z^2 to the
limit by strong L^2 convergence, and the negative part likewise,
assembles the lost positive entropy as at most 2s(E_* - ||Du||_2^2).
The normalization terms vanish because ||u||_2 = ||u_j||_2 = 1. QED.
The coefficient 2s is where "escape into the Gaussian tails costs
entropy at rate at most 2s" is made precise. The next subsection uses
this defect to prove that a maximizer of the entropy quotient exists.
6.5 Existence of an optimizer (strict branch).
The strict branch is the case where the entropy constant strictly beats
both the Poincare constant and the Gaussian-regularization scale. There
the entropy quotient is attained.
Lemma 9 (Existence of an optimizer). Let nu = mu * gamma_s, with s > 0
and mu compactly supported. Suppose R = R(nu) and P = R^P(nu) satisfy
infinity > R > max(P, s). Then there is a smooth strictly positive
nonconstant f with
int f^2 dnu = 1, Ent_nu(f^2) = 2 R int |Df|^2 dnu,
and f satisfies the logarithmic Euler equation
A_nu f = lambda f log f, lambda = R^{-1},
first distributionally and then classically.
Proof of existence and the Euler equation. Take nonnegative unit-norm
maximizing tests f_j. Replacing a test by its absolute value preserves
entropy and energy, so this loses nothing. The defective inequality of
Lemma 8, applied with s + eps < R, bounds the energies ||D f_j||_2.
First exclude energies tending to zero; this is the only place the
strict inequality R > P is used. Suppose eps_j^2 = ||D f_j||_2^2 -> 0,
and write c_j = E_nu f_j, h_j = (f_j - c_j)/eps_j. Poincare gives
||h_j||_2^2 <= P, while ||D h_j||_2^2 = 1 and 1 - c_j^2 = eps_j^2
||h_j||_2^2, so c_j -> 1. By compactness, after extracting, h_j -> h in
L^2 with E h = 0, ||Dh||_2^2 <= 1, and ||h||_2^2 <= P ||Dh||_2^2.
For c > 0, eps > 0 and z >= -c/eps define
Q_{c,eps}(z) = eps^{-2} [ (c+eps z)^2 log((c+eps z)^2/c^2)
- (c+eps z)^2 + c^2 ].
Normalization and -log c_j^2 - 1 + c_j^2 >= 0 give
eps_j^{-2} Ent(f_j^2) <= int Q_{c_j,eps_j}(h_j) dnu.
The scalar estimate we need: for 1/2 <= c <= 1 and 0 < eps <= 1/2,
0 <= Q_{c,eps}(z) <= (2+2log2) z^2 + z^2 log_+ z^2,
Q_{c,eps}(z) -> 2 z^2 uniformly on bounded z-intervals.
To verify, put t = eps z / c, so Q = z^2 B(t) with
B(t) = 2 + 4 int_0^1 (1-r) log(1 + r t) dr, -1 <= t < infinity. For
t <= 0, B(t) <= 2; for t >= 0, t <= z gives
B(t) <= 2 + 2 log(1+z) <= 2 + 2log2 + log_+ z^2.
Now use the general tail bound: the tail argument in Lemma 8 never
used unit norm until its last line, so for any bounded form sequence
u_j -> u in L^2 with energies -> E_*,
lim_{K} limsup_j int_{|u_j|>K} u_j^2 log_+ u_j^2 dnu
<= 2s (E_* - ||Du||_2^2).
Apply the compact convergence in the scalar estimate on |h_j| <= K and
the upper bound on the complement. Strong L^2 convergence removes the
quadratic tail as K -> infinity; the general tail bound controls the
entropy tail. Therefore
limsup_j eps_j^{-2} Ent(f_j^2)
<= 2||h||_2^2 + 2s(1 - ||Dh||_2^2) <= 2 max(P, s) < 2R,
contradicting the maximizing property. So energies do not vanish.
Hence we may extract f_j -> f strongly in L^2 with
||D f_j||_2^2 -> E_* > 0. Put E = ||Df||_2^2. The entropy defect and
the sharp inequality give
2R E_* <= Ent(f^2) + 2s(E_* - E) <= 2R E + 2s(E_* - E).
Since R > s and E <= E_*, equality forces E = E_* > 0. Thus f is a
nonnegative, nonconstant optimizer.
The homogeneous functional Ent(g^2) - 2R ||Dg||_2^2 is nonpositive and
is zero at f. Its derivative along every compact smooth variation zeta
is
4 int f log f zeta dnu - 4R int Df.Dzeta dnu = 0.
The entropy normalization term cancels the additional linear term, and
differentiation at zeros is legitimate because z^2 log z^2 has zero
derivative at zero. This is the Euler equation A_nu f = lambda f log f
in distributional form. Smoothness and strict positivity are handled
next. QED.
So in the strict branch we have a bona fide optimizer f, and w = D log
f^2 must be a nearly first eigenfunction. Turning the Euler equation
into that statement, and proving f is smooth and positive, is the rest
of the section.
6.6 Regularity and strict positivity of the optimizer.
We have a nonnegative optimizer satisfying the weak Euler equation. To
use its logarithmic derivative we need local regularity and strict
positivity.
Regularity. Write rho = exp(-V) for the smooth density of nu, set
v = exp(-V/2) f >= 0. Differentiating the weak equation directly gives
-Delta v = lambda v log v
+ ( lambda V/2 - |DV|^2/4 + Delta V/2 ) v.
The conjugating factor and its reciprocal are bounded on each fixed
ball. Since |z log z| <= C_delta (1 + z^{1+delta}) for z >= 0, local
L^p integrability of v puts the right side in L^q_loc with
q = p/(1+delta) > 1. Localizing the right side on a larger ball and
taking its Newtonian potential u solving -Delta u = F, the interior L^q
estimate gives u in W^{2,q}_loc; the difference v - u is
distributionally harmonic, hence smooth by Weyl lemma. Then Sobolev
embedding improves the integrability of v step by step: before
boundedness is reached, the reciprocal integrability exponent 1/p
improves to (1+delta)/p - 2/n, a gain. After finitely many steps q >
n/2, so v, hence f, is locally bounded and Holder continuous; the
local Schauder estimate then gives f in C^{2,alpha}_loc. In dimension
one the same conclusion follows by twice integrating. Every constant
here concerns one fixed law and dimension and never enters a uniform
estimate along the sequence.
Positivity. Suppose f(x_0) = 0. Choose a ball B about x_0 on which
0 <= f <= b < 1 on its closure. Start the nu-diffusion of Lemma 3 at
x_0 and stop it at the exit time tau from B. Put Z_t = f(Y_{t and tau}),
m(t) = E Z_t, g(z) = -lambda z log z. The stopped Ito formula gives,
for almost every t,
m'(t) = E[ 1_{t<tau} g(Z_t) ] <= E g(Z_t) <= g(m(t)), m(0) = 0,
because g is nonnegative and concave on [0,b] and g(0)=0. Also m' >= 0.
If m(t_1) > 0 with t_1 > 0, and t_a is the first time m = a, then for
0 < a < m(t_1),
int_a^{m(t_1)} dz/( -lambda z log z ) <= t_1 - t_a <= t_1,
contradicting the divergence of the integral as a -> 0. Hence m = 0.
The killed nondegenerate diffusion reaches every open subset of B with
positive probability (follow a smooth interior path and require the
Brownian motion to stay in a thin tube), so f vanishes on all of B, and
its zero set is both open and closed. Since R^n is connected and f is
nonzero, f must be strictly positive. Smooth elliptical bootstrap then
makes f smooth everywhere. QED.
6.7 Change of reference at an optimizer.
An optimizer supplies a second probability with its own LSI, and no
convexity of that second probability is needed.
Lemma 10 (Change of reference). Under Lemma 9, suppose in addition that
nu = exp(-V) is log-concave with bounded curvature K = D^2 V. Define
eta = f^2 nu, h = log f^2, w = D h, I = E_eta |w|^2.
Then I = 2 lambda H(eta | nu) > 0, R(eta) <= 1/lambda, each component
of w lies in the domain of A_eta, and
A_nu h = lambda h + (1/2)|w|^2, (i)
A_eta w = (lambda Id - K) w. (ii)
Moreover E_eta <w, K w> <= lambda |E_eta w|^2. (iii)
Proof. Since w = 2 Df/f, I = 4 ||Df||_{L^2(nu)}^2; optimizer equality
gives I = 2 lambda H(eta|nu) > 0. The chain rule applied to the Euler
equation gives (i). Commuting one derivative with A_nu gives
A_nu w = D(A_nu h) - K w = lambda w + (Dw) w - K w.
Since A_eta = A_nu - w.D and Dw is symmetric, (ii) follows
distributionally. Its right side is in L^2(eta) because K is bounded and
w in L^2(eta), so Lemma 3 puts w in the operator domain; no unproved
higher integrability has been used.
For the inequality, let p = g^2 eta have unit mass with g smooth compact.
Direct expansion and integration by parts give
Ical(p|nu) = Ical(p|eta) + int g^2 |w|^2 deta + 2 int D(g^2).w deta
= Ical(p|eta) + int g^2 (|w|^2 + 2 A_eta h) deta
= Ical(p|eta) + 2 lambda int h dp.
On the other hand H(p|nu) = H(p|eta) + int h dp. Subtracting
2 lambda times this gives the exact deficit identity
Ical(p|nu) - 2 lambda H(p|nu) = Ical(p|eta) - 2 lambda H(p|eta).
Apply the nu-inequality to the compact smooth function g f: its deficit
is nonnegative, so the right side is nonnegative. Homogeneity and
Sobolev approximation give R(eta) <= 1/lambda. Finally, the domain
identity gives E_eta|Dw|^2 = lambda I - E_eta<w,Kw>, while Poincare for
eta gives the lower bound lambda(I - |E_eta w|^2). Subtract to obtain
(iii). QED.
The lemma does two things at once. It transfers a logarithmic Sobolev
inequality from nu to the tilted law eta = f^2 nu (so eta inherits the
good entropy constant), and it writes the score w as an approximate
first eigenfunction of eta's diffusion. Everything after this point
happens under eta.
6.8 Translation gain: uniform control of means.
The critical law eta has a near-first-eigenfield w, but we also need to
know that its mean and its curvature-action are small. The tool is a
stability estimate for the LSI under translating a probability.
Lemma 11 (Translation gain). Let mu satisfy the normalized
counterexample conditions of Lemma 6 with a <= 1. Fix a compact
interval J inside (0, 1/2). There is a constant C_J, independent of the
dimension and of the support radius of mu, with the following property.
For s in J, put lambda = R_s^{-1}. If sigma is any probability with
D_0 = H(sigma | mu_s) < infinity, F_0 = Ical(sigma | mu_s) < infinity,
and p = E_sigma Y is its mean, then
F_0 - 2 lambda D_0 >= ( s^{-2} - lambda/s ) |p|^2
- C_J sqrt(a) ( F_0 + D_0 ).
Proof. The centered linear moment generating functions under mu_s obey
log E_{mu_s} exp(v.Y) <= C_J |v|^2,
log E_{mu_s} exp(v.b_s(Y)) <= C a^2 |v|^2,
the first by independent Gaussian addition and the moment bound of
Lemma 6, the second by conditional Jensen. Entropy duality, optimized
over the scalar multiple of each unit vector, gives
|p| <= C_J sqrt(D_0), |E_sigma b_s(Y)| <= C a sqrt(D_0).
Compact support of X makes E_{mu_s} exp(c|Y|^2) finite for some c > 0,
so all these expectations exist; by duality E_sigma |Y|^2 < infinity.
The latter is used only for integrability, not for a dimension bound.
Let sigma_u be the translate of sigma by -u p, 0 <= u <= 1, and let
D_u = H(sigma_u | mu_s). Change of variables in entropy and the
Hessian bound give
D_u = D_0 + E_sigma[ V_s(Y - up) - V_s(Y) ]
<= D_0 - u|p|^2/s + u p.E_sigma b_s(Y)/s + u^2 |p|^2/(2s)
<= C_J D_0.
Thus the mean estimate applied to sigma_u gives
|E_sigma b_s(Y - up)| <= C_J a sqrt(D_0) uniformly in u. Differentiating
the exact entropy expression and integrating gives
D_1 = D_0 - |p|^2/(2s) + O_J(a D_0).
Its derivative is -(1-u)|p|^2/s + p.E_sigma b_s(Y - up)/s.
Write r(Y) = D log( dsigma / dmu_s )(Y) in the weak square-root sense.
Its mean is
E_sigma r = E_sigma D V_s = ( p - E_sigma b_s ) / s.
To justify this for finite Fisher information, the Lebesgue density q of
sigma has weak derivative Dq = q(r - D V_s) in L^1: r is square
integrable, and D V_s has linear growth. Integrating against expanding
cutoffs gives int Dq = 0.
In moving coordinates Y, the score of sigma_1 relative to mu_s is
r(Y) - p/s + d(Y)/s with d(Y) = b_s(Y) - b_s(Y - p). Since
0 <= D b_s <= Id, integration along the segment gives
|d(Y)|^2 <= int_0^1 |M_s(Y - up) p|^2 du <= p.d(Y). The mean estimates
at the two endpoints therefore imply
E_sigma |d|^2 <= C_J a D_0.
Expanding the score square and using the mean identity,
E_sigma |r - p/s|^2 = F_0 - |p|^2/s^2 + 2 p.E_sigma b_s/s^2.
The remaining cross term is bounded by C_J sqrt(a D_0)(sqrt(F_0) +
sqrt(D_0)), and the d-square term by C_J a D_0. Therefore the translated
information is finite and
F_1 = F_0 - |p|^2/s^2 + O_J( sqrt(a) (F_0 + D_0) ).
These calculations use only the weak derivatives already justified, so
they apply when the density of sigma vanishes. Apply the LSI to sigma_1,
F_1 >= 2 lambda D_1, and combine with the two translated displays. Since
2/3 <= lambda <= 1 on J, the result is the claim. QED.
The lemma says: any probability close to mu_s in entropy and Fisher
information cannot have a large mean. A nonzero mean would force a
strictly positive energy surplus, at least (s^{-2} - lambda/s)|p|^2, and
that surplus is controlled up to sqrt(a) times the sizes involved. Since
a -> 0, means are forced to zero.
6.9 The critical fields in both branches.
The construction of Section 6.7 leaves a near-first eigenfunction in
the strict branch; the gap branch produces its own. Both satisfy the
same structural identities with the same uniform error estimates.
Lemma 12 (Critical fields). Fix a compact interval J inside (0, 1/2)
and s in J. Put lambda = 1/R_s. There exist a smooth positive
probability eta and a smooth vector field w such that
0 < I := E_eta |w|^2 < infinity,
R(eta) <= lambda^{-1}, w in Dom(A_eta)^n,
A_eta w = (lambda Id - K_s) w,
|E_eta w|^2 + E_eta |K_s w|^2 <= C_J sqrt(a) I,
E_eta <w, K_s w> <= lambda |E_eta w|^2.
In the gap branch sqrt(a) may be replaced by a^2. The coordinates Y
belong to Dom(A_eta) in both branches. The choices are:
strict branch: eta = f^2 mu_s, w = D log f^2, I = 2 lambda H(eta | mu_s);
gap branch: eta = mu_s, w = D varphi,
where varphi is a centered eigenfunction of A_{mu_s} with eigenvalue
lambda, and I = lambda E_{mu_s} varphi^2.
Proof. Strict branch. Lemma 10 supplies the structural identities and
the curvature-mean inequality. Apply Lemma 11 to sigma = eta with
F_0 = I and D_0 = I/(2 lambda). The deficit vanishes, so its conclusion
reads
(s^{-2} - lambda/s) |p|^2 <= C_J sqrt(a) I,
where p = E_eta Y. Because s^{-2} - lambda/s >= 2 on J, and
|E_eta b_s| <= C_J a sqrt(I) comes from the same proof, the score
identity E_eta w = (p - E_eta b_s)/s yields |E_eta w|^2 <= C_J sqrt(a) I.
Now 0 <= K_s <= s^{-1} Id, so K_s^2 <= s^{-1} K_s, and the
curvature-mean inequality gives
E_eta |K_s w|^2 <= s^{-1} E_eta <w, K_s w> <= (lambda/s) |E_eta w|^2.
The two displayed bounds are the first line of uniform smallness.
Gap branch. Compact form embedding supplies a minimizer of the
Rayleigh quotient for A_{mu_s} among mean-zero unit-norm functions.
Its positive eigenvalue is 1 / R_s^P = lambda. The nullspace is the
constants, so the quotient has a positive gap. Interior elliptic
regularity makes its eigenfunction varphi smooth. Commuting one
derivative gives
A_{mu_s} D varphi = (lambda Id - K_s) D varphi.
The right side is in L^2(mu_s), so Lemma 3 puts D varphi in the
operator domain. Pairing this display with varphi and applying
Poincare at mu_s reproduces the curvature-mean inequality
E_eta <w, K_s w> <= lambda |E_eta w|^2.
The coordinate function Y also sits in the operator domain, because
both Y and A_{mu_s} Y = (Y - b_s)/s lie in L^2(mu_s). Pairing
A_{mu_s} varphi = lambda varphi against Y, using self-adjointness,
gives
E_eta w = lambda E[varphi Y]
= (E[varphi Y] - E[varphi b_s]) / s,
(1 - lambda s) E[varphi Y] = E[varphi b_s].
Conditional Jensen and Lemma 6 give E_{mu_s} b_s b_s^* <=
E_mu X X^* <= C a^2 Id. For every unit vector v, Cauchy-Schwarz
bounds |E[varphi v.b_s]| by C a ||varphi||_2, with no factor from
dimension. Since 1 - lambda s >= 1/2 and I = lambda ||varphi||_2^2, the
display gives |E w| <= C_J a sqrt(I), and K_s^2 <= s^{-1} K_s
finishes the uniform smallness of K_s w.
Coordinate assertion in the strict branch. H(eta | mu_s) < infinity
together with the Gaussian tails of mu_s implies E_eta |Y|^2 <
infinity. The distributional identity
A_eta Y = (Y - b_s)/s - w
has an L^2(eta) right side, because b_s is bounded for the fixed law
and w lies in L^2(eta). Lemma 3 applies. QED.
6.10 How the regularized constants converge to one.
Proposition 13. For the sequence of normalized counterexamples,
R_s -> 1 for every 0 < s < 1/2.
At each differentiability point of R_s, the fields of Lemma 12 satisfy
the contact identity
R_s' = R_s * ( E_eta <w, K_s w> / I ).
In particular 0 <= R_s' <= C_J sqrt(a) almost everywhere on each fixed
compact J.
Proof. Fix s, and let Phi_{s,t} be the heat continuity flow of Lemma 7.
Transport a form-domain test g by g_t = g o Phi_{s,t}^{-1}. Its
distribution under mu_t is fixed, so entropy and variance are constant.
The Jacobian equation d DPhi / dt = K_t DPhi / 2 gives
(d/dt at t = s) of int |D g_t|^2 dmu_t
= - int <Dg, K_s Dg> dmu_s.
For compact smooth g this follows by differentiating the
inverse-transpose Jacobian inside the gradient. A general form-domain
g follows by Sobolev approximation: flow derivatives and their time
derivatives are uniformly bounded on a compact time interval, and the
derivative of the integrand is dominated by C |Dg|^2. Dominated
convergence proves the identity without smoothness or compact support.
Strict branch. Take g = f and Q(t) = Ent_{mu_t}(f_t^2) /
(2 int |D f_t|^2 dmu_t). The denominator is positive near s, and
Q(t) <= R_t for both signs of t - s, with Q(s) = R_s. At a
differentiability point of R_s the two-sided contact gives
Q'(s) = R_s'. The energy identity and w = 2 Df / f give
R_s' = R_s E<w, K_s w> / I.
Gap branch. Take g = varphi and Q(t) = Var_{mu_t}(varphi_t) /
int |D varphi_t|^2 dmu_t. Now Q(t) <= R_t^P <= R_t and
Q(s) = R_s^P = R_s. The same contact argument gives the identical
derivative formula with w = D varphi.
Lemma 12 bounds the derivative by C_J sqrt(a) in either branch, with
constants independent of which branch holds at each time. Lipschitz
continuity makes R_s absolutely continuous. For fixed
0 < delta < s < 1/2, integrating on [delta, s],
1 <= R_s
<= R_delta + C_{[delta, s]} sqrt(a) (s - delta)
<= 1 + delta + C_{[delta, s]} sqrt(a) (s - delta).
Take the sequence limit first, then delta -> 0. QED.
6.11 The full critical sequence.
Proposition 14. If Theorem 1 fails, there is a sequence of dimensions
and centered compactly supported log-concave laws mu satisfying the
normalization of Lemma 6. Fix s = 1/8, form mu_s, b_s, M_s, K_s, and
put lambda = 1/R_s. After passage to a subsequence there exist smooth
positive probabilities eta and smooth vector fields w such that, with
A = A_eta and I = E_eta |w|^2,
lambda -> 1, R(eta) <= lambda^{-1}, 0 < I < infinity,
w in Dom(A)^n, A w = (lambda Id - K_s) w,
||(A - 1) w||_{L^2(eta)} + |E_eta w| = o(sqrt I),
E_eta |K_s w|^2 = o(I), E_eta |Dw|^2 = (1 + o(1)) I.
The spectral gap of A is at least lambda. Each coordinate function
belongs to Dom(A). Either of the following holds throughout the
subsequence.
(i) R_s > R_s^P: eta = f^2 mu_s for the positive unit-norm optimizer,
w = D log f^2, I = 2 lambda H(eta | mu_s). The exact deficit
identity of Lemma 10 holds for compact square-root tests
relative to eta, and A Y = (Y - b_s)/s - w.
(ii) R_s = R_s^P: eta = mu_s, A varphi = lambda varphi,
E_eta varphi = 0, w = D varphi, I = lambda E varphi^2.
Coefficients may be scaled positively so that I -> 0.
All o-terms refer to the sequence, with all finite auxiliary parameters
held fixed. No relation between dimension, I, and the rate of
convergence is asserted or required.
Proof. Choose an infinite subsequence with a fixed branch at s = 1/8.
Lemma 12 gives the structural identities, together with the curvature
and mean bounds. Proposition 13 gives lambda -> 1, hence
(A - 1) w = (lambda - 1) w - K_s w = o_{L^2(eta)}(sqrt I).
The energy pairing yields
E|Dw|^2 = lambda I - E<w, K_s w> = (1 + o(1)) I,
and Poincare with constant lambda^{-1} is the asserted spectral gap.
The coordinate assertion and the branch-specific identities are
already proved. In the gap branch every estimate is homogeneous in
varphi, so positive rescaling enforces I -> 0 while leaving each
normalized claim intact. QED.
7. THE PREDICTION CURVE
The critical sequence of Proposition 14 provides the following data.
Fix s = 1/8 and choose one of its two subsequences, suppressing the
sequence index. Then
lambda -> 1, R(eta) <= 1/lambda, A = A_eta,
I = E_eta |w|^2 > 0, w in Dom(A)^n,
||(A - 1) w||_{L^2(eta)} + |E_eta w| = o(sqrt I). (7.1)
Strict branch: eta = f^2 mu_s, w = D log f^2, 2 H(eta | mu_s) = I / lambda.
Gap branch: eta = mu_s, w = D varphi, A_{mu_s} varphi = lambda varphi,
E_{mu_s} varphi = 0, I = lambda E_{mu_s} varphi^2.
Every parameter T, M, r, epsilon is fixed before the sequence limit.
7.1 Efficient Gaussian probes (strict branch).
Let Y_0 ~ eta and let G' be an independent standard Gaussian vector.
Define the total observation energy
L_T = sqrt(T) w(Y_0) + G', d(T) = H(Law(L_T) | gamma), (7.2)
where gamma is the standard Gaussian on the same Euclidean space. If w
were an exact first eigenfunction of A, d(T) would vanish identically in
T. The next lemma says d(T) is small.
Lemma 15 (Efficient probes). In the strict branch, for every finite M,
sup_{0 <= T <= M} d(T) = o(I). (7.3)
Proof. Run the stationary eta-diffusion with independent Brownian noise
and semigroup P_u = e^{-uA}. Proposition 14 combined with the
coordinate-generator identity of Lemma 12 gives
-A Y = -Y/s + w + b_s/s. (7.4)
Consequently E_eta |Y_0|^2 < infinity. The LSI of eta gives
E_eta exp(v.(Y_0 - E Y_0)) <= exp(|v|^2 / (2 lambda)).
Integrating against a Gaussian v shows E_eta exp(c' |Y_0|^2) < infinity
for some c' > 0 depending on the individual law. This exponential-square
moment is the hypothesis of Lemma 16.
Fix T <= M. Define
J_u = MI(L_T; Y_u),
F_u = E Ical(Law(Y_u | L_T) | eta),
Q_u = E |E[Y_u | L_T]|^2.
The Gaussian observation identity of Lemma 5 gives
J_0 = T I / 2 - d(T). Entropy dissipation yields -dJ_u >= F_u du, so
int_0^1 F_u du <= J_0. For sigma_{u,l} = Law(Y_u | L_T = l), the chain
rule and the strict-normalization identity give
E H(sigma_{u,L_T} | mu_s) = J_u + I/(2 lambda),
E Ical(sigma_{u,L_T} | mu_s) = F_u + I.
For the second identity, the conditional relative score with respect to
mu_s is the sum of w and the conditional relative score with respect to
eta. The latter scores have conditional mean zero at each Y_u, so their
contribution vanishes weakly, since the conditional likelihoods average
to one under the marginal of Y_u.
Apply Lemma 11 to sigma_{u,l} and average over l. Since
s^{-2} - lambda/s >= 2, on [0,1]
-dJ_u >= (2 lambda J_u + c_s Q_u) du - d E_u, E([0,1]) = o_M(I), (7.5)
with c_s > 0 and E a nonnegative error measure. The error in Lemma 11
is bounded by a quantity vanishing along the sequence times
F_u + I + J_u + I/(2 lambda), whose integral is O_M(I). The measure
formulation avoids differentiability of J_u.
The spectral theorem and (7.1) give
sup_{0 <= u <= 1} ||P_u w - e^{-u} w||_2
<= ||(A - 1) w||_2 = o(sqrt I). (7.6)
Reversibility gives E[L_T | Y_u] = sqrt(T) P_u w(Y_u). The entropy
duality of Lemma 5 with a linear test function gives
J_u + d(T) = E H(Law(L_T | Y_u) | gamma)
>= (T/2) ||P_u w||_2^2
= (T I / 2) e^{-2u} + o_M(I).
Integrating (7.5) with factor e^{2 lambda u} and comparing at u = 1,
int_0^1 Q_u du <= C_s d(T) + o_M(I). (7.7)
The difference e^{-2 lambda} - e^{-2} contributes only o_M(I).
Bound the predicted position of b_s. By conditional Jensen and Lemma 6,
log E_{mu_s} exp(v.b_s) <= C a^2 |v|^2.
The entropy variational formula, optimized over v, gives
E |E[b_s(Y_v) | L_T]|^2 <= C a^2 (J_v + I/(2 lambda)) = o_M(I).
Condition (7.4) on L_T and solve by variation of constants. By (7.6)
and contraction of conditional expectations,
E[Y_u | L_T] = e^{-u/s} E[Y_0 | L_T]
+ ((e^{-u} - e^{-u/s}) / (1/s - 1)) E[w(Y_0) | L_T]
+ o_{L^2, M}(sqrt I), (7.8)
uniformly for 0 <= u <= 1. The two scalar coefficient functions are
linearly independent on [0,1]: the first is nonzero at 0, and the
second vanishes at 0 and is nonzero for u > 0. Their 2 x 2 Gram matrix
is positive definite. Applying its smallest-eigenvalue bound to each
coordinate and summing, (7.7) and (7.8) show
E |E[w(Y_0) | L_T]|^2 <= C_s d(T) + o_M(I).
By Lemma 5 the left side equals 2 d'(T) for almost every T. The error
is uniform for T in [0, M], and d(0) = 0. Integrating the resulting
differential inequality proves (7.3). QED.
7.2 Entropy loss under a Gaussian observation.
The next comparison is the lower-bound engine of the prediction curve.
If heat smoothing removes only a little of the initial relative entropy,
then the score's best predictor must already be large. The comparison
discards at each new observation the direction in which the two
candidate posterior means disagree. In continuous-time stochastic
localization, Eldan-Koehler-Zeitouni and Chen-Eldan control this cost
via projections chosen to preserve a selected function. The argument
below is direct: it uses finite Gaussian observations with singular
projections, and takes independent copies of the entire experiment.
Lemma 16 (Entropy loss). Let p, q be smooth strictly positive
probability densities on R^d, with q log-concave. Suppose
H_0 = H(p | q) < infinity, w = D log(p/q), I = E_p |w|^2 < infinity,
E_p exp(c |Y|^2) < infinity for some c > 0,
where Y ~ p. Let G be a standard Gaussian independent of Y. For r > 0
put U_r = Y + sqrt(r) G, p_r = p * gamma_r, q_r = q * gamma_r,
wbar_r(U_r) = E_p[w(Y) | U_r], H_r = H(p_r | q_r).
Then for every z > 0,
2 (H_0 - H_z) <= int_0^z E_p |wbar_r(U_r)|^2 dr. (7.9)
Proof. Setup. Put b = 1/z. Take N independent copies of Y and a
Gaussian observation mesh of size m; put delta = u/m. The three limits
(mesh, copies, extra precision) are taken in that order.
Additional observations and their entropy cost. Write
Y = (Y_1,...,Y_N) and W = (w(Y_1),...,w(Y_N)). The two candidate
priors are p^N and q^N; expectations below are with respect to p^N.
First observe Z_0 = sqrt(b) Y + G_0. By the chain rule, the posterior
relative entropy averaged over this observation equals N(H_0 - H_z).
Fix u > 0 and mesh m, with delta = u/m. For i = 0,...,m-1 let H_i be
the retained history up to step i. Compute the two posterior means
under p and q given H_i, and let P_i be the orthogonal projection onto
their span, with P_i = 0 when the span is trivial. Retain only
(1 - P_i) Z_{i+1}, Z_{i+1} = sqrt(delta) Y + G_{i+1},
with all Gaussian noises independent and the same rule applied under
both priors. Integral-formula versions of the posterior means make the
rule measurable.
For each realized retained history, the likelihood of Y is Gaussian
with precision matrix
B = b Id + delta sum_{i=0}^{m-1} (1 - P_i) >= b Id. (7.10)
No extra factor enters from the choices of P_i, since each choice was
determined by the retained past; the factor is therefore the same under
both priors. The posterior relative score of p against q is still W.
Bound the entropy lost in the additional observations. Condition on a
retained history at one step and put Q = 1 - P_i. Under either prior
the projected posterior means agree. Subtract their common mean and
let X_c denote the resulting centered projected signal in ran Q. The
base posterior has curvature at least b Id. Its centered linear moment
bound and Gaussian integration give
E_q [exp(kappa |X_c|^2) | H_i] <= (1 - 2 kappa/b)^{-Nd/2},
0 < 2 kappa < b.
For the alternative prior write a_i = E_p[Y | H_i]. For sufficiently
small kappa, conditional Jensen gives
E_p[exp(kappa |Q(Y - a_i)|^2) | H_i]
<= exp(2 kappa |a_i|^2) E_p[exp(2 kappa |Y|^2) | H_i]
<= (E_p[exp(2 kappa |Y|^2) | H_i])^2.
Squaring and applying conditional Jensen once more bounds the average
of this square over H_i by E_p exp(8 kappa |Y|^2) < infinity. The
constants are independent of the mesh and of the adaptive choices.
Let h_p, h_q be the output densities of sqrt(delta) X_c + G in ran Q,
and let phi be the standard Gaussian density there. Centering and
Jensen give h_q / phi >= exp(-delta Nd/(2 b)) pointwise. For either
conditional signal, integrating the Gaussian kernel produces
|| h/phi - 1 ||_{L^2(phi)}^2 = E[exp(delta X_c . X_c') | H_i] - 1,
where X_c' is a conditionally independent copy. The linear term
vanishes because X_c is centered. The inequality
|exp(t) - 1 - t| <= (t^2/2) exp(|t|), together with
|X_c . X_c'| <= (|X_c|^2 + |X_c'|^2)/2, bounds the last expression by
C delta^2 times a squared small conditional exponential-square moment.
The preceding bounds therefore give
E H(h_p | h_q) <= E chi^2(h_p | h_q) <= C_{N,b,p} delta^2.
This includes the zero-dimensional observed subspace, where the loss is
zero. Summing the m steps loses O_{N,b,p}(u delta).
Apply the anisotropic curvature inequality of Lemma 1 to the final
posterior pair: on each realized history the covariance of the retained
observations has curvature at least B. For fixed N and u,
2 N (H_0 - H_z) - o_{delta -> 0}(1) <= E W^* B^{-1} W. (7.11)
Comparison with full observations. Work in the Hilbert space of
square-integrable vector fields on the signal and all full observations
Z_0,...,Z_m. Then ||W||_2^2 = N I. Completing the square pointwise in v,
b ||W||_2^2 - b^2 E W^* B^{-1} W
= min_v [ b ||W - v||_2^2
+ delta sum_{i=0}^{m-1} ||(1 - P_i) v||_2^2 ]. (7.12)
The minimizer is v_* = b B^{-1} W, so ||v_*||_2 <= ||W||_2.
Let F_i = sigma(Z_0,...,Z_i) and e_i = E[. | F_i]. Replace 1 - P_i by
1 - e_i in (7.12) and let M_0 be the resulting minimum. This modified
minimum has the explicit minimizer
v_0 = b (b + delta sum_{i=0}^{m-1} (1 - e_i))^{-1} W.
Use the orthogonal decomposition into increments of the nested
sigma-fields. Put alpha_k = b / (b + k delta) for 0 <= k <= m. On
ran e_0 the multiplier is 1; on ran(e_k - e_{k-1}) it is alpha_k; on
ran(1 - e_{m-1}) it is alpha_m. Therefore
v_0 = alpha_m W + sum_{k=0}^{m-1} (alpha_k - alpha_{k+1}) e_k W. (7.13)
All coefficients are nonnegative and sum to one. Put g_i = (1 - e_i) v_0.
Equation (7.13) gives
g_i = alpha_m (W - e_i W)
+ sum_{k=i+1}^{m-1} (alpha_k - alpha_{k+1})(e_k W - e_i W). (7.14)
The weights in this formula sum to alpha_{i+1} <= 1.
Independence across copies, boundedness of the retained histories, and
the conditional covariance structure of the observation arrays put
Z_{i,j} := tr Cov(g_{i,j} | F_i)
<= E[|W_j - e_i W_j|^2 | F_i]
<= E[|W_j|^2 | F_i]. (7.15)
For every term with k > i, the tower property bounds the conditional
squared mean by E[|W_j - e_i W_j|^2 | F_i], and nonnegative weights of
total at most one preserve this bound.
The right sides in (7.15) are uniformly integrable over the sigma-
fields appearing. Concretely, if Z >= 0 is integrable and M = E[Z | F],
E[M 1_{M > K}] <= 2 E[Z 1_{Z > K/2}]. (7.16)
Indeed M <= K/2 + E[Z 1_{Z > K/2} | F], and on {M > K} the final
conditional expectation is at least M/2. Since P_i is F_i-measurable
and rank at most one,
E |P_i g_i|^2 <= E max_{1 <= j <= N} Z_{i,j}.
The maximum is at most K + sum_j Z_{i,j} 1_{Z_{i,j} > K}. Equations
(7.15) and (7.16) therefore give, uniformly in i, m, delta,
E |P_i g_i|^2 <= N q_N,
q_N := inf_{K > 0} { K/N + 2 E_p [|w(Y)|^2 1_{|w(Y)|^2 > K/2}] } -> 0. (7.17)
No moment of the score beyond its second moment enters. The bound
permits P_i to depend adaptively on all copies.
A dual test compares the original minimum with M_0. For every v,
||(1 - P_i) v||_2^2 >= 2 <v, (1 - P_i) g_i> - ||g_i||_2^2.
The normal equation for v_0 is b (W - v_0) = delta sum_i g_i. So the
minimum over v of
b ||W - v||_2^2 + 2 delta sum_i <v, g_i> - delta sum_i ||g_i||_2^2
attains the value M_0 at v_0. Applying the linear bound at v_* in
(7.12) and using (7.17),
min >= M_0 - 2 delta sum_i ||v_*||_2 ||P_i g_i||_2
>= M_0 - 2 u N sqrt{I q_N}. (7.18)
The prediction integral. The same nested decomposition gives
M_0 / N
= sum_{i=0}^{m-1}
b^2 delta / ((b + i delta)(b + (i+1) delta))
* (I - ||e_i W_1||_2^2). (7.19)
A component first seen after j increments pays b j delta / (b + j delta),
and this equals sum_{i=0}^{j-1} b^2 delta / ((b + i delta)(b + (i+1) delta)).
Full observations through step i constitute a Gaussian sufficient
statistic of precision b + i delta; completing the likelihood square
proves sufficiency directly. Consequently
||e_i W_1||_2^2 = E_p |wbar_{1/(b+i delta)}(U_{1/(b+i delta)})|^2.
This squared prediction norm is nondecreasing in precision: a weaker
observation is obtained by convolving with fresh Gaussian noise. The
norms are bounded by I. So (7.19) converges as m -> infinity to
b^2 int_0^u (I - E_p |wbar_{1/(b+t)}(U_{1/(b+t)})|^2) / (b + t)^2 dt.
Limits. Take the mesh limit first, at fixed N, u. The entropy error in
(7.11) vanishes, and the error in (7.18) remains uniform in the mesh.
Divide by N. Let N -> infinity, using q_N -> 0. Combine (7.11) and
(7.12):
2(H_0 - H_z) <= I/b
- int_0^u (I - E_p |wbar_{1/(b+t)}(U_{1/(b+t)})|^2) / (b + t)^2 dt
= I/(b+u) + int_{1/(b+u)}^{1/b} E_p |wbar_r(U_r)|^2 dr.
Finally take u -> infinity. Since 1/b = z and the integrand is bounded
by I, this is (7.9). QED.
7.3 The prediction curve.
For Y_0 ~ eta and independent Gaussian noise define
U_r = Y_0 + sqrt(r) G,
wbar_r(U_r) = E[w(Y_0) | U_r],
m(r) = (1/I) E |wbar_r(U_r)|^2.
In the strict branch, let L_T be the labeled observation of Lemma 15.
It is constructed with fresh Gaussian noise independent of (Y_0, G), so
L_T and U_r are conditionally independent given Y_0. Define
p_T(r) = (1/I) E |E[w(Y_0) | U_r, L_T]|^2, p_0(r) = m(r).
Proposition 17 (Prediction curve). In both branches, for every fixed
0 < r < 1/4,
m(r) -> e^{-r}. (7.20)
In the strict branch, p_T(r) -> e^{-r} for every fixed finite T >= 0.
In the gap branch the eigenfunction may be rescaled to make I -> 0
without changing (7.20) or (7.1).
Proof. For 0 <= r < 1/4 put theta(r) = exp(-int_0^r dt / R_{s+t}).
Proposition 13 gives theta(r) = e^{-r} + o(1) uniformly on
compact intervals. The common heat flow dissipates relative entropy
at one half of the relative Fisher information. Its entropy relative to
mu_{s+r} is therefore at most theta(r) times its initial entropy, by
the LSI at the contemporaneous base law. The same assertion holds for
conditional laws after averaging.
Strict branch. Write H_r = H(Law(U_r) | mu_{s+r}) and
H_0 = I/(2 lambda). The entropy of U_r given L_T, relative to mu_{s+r}
and averaged over the label, is at most theta(r)(H_0 + TI/2), since
initially it is H_0 + MI(Y_0; L_T). Chain rule and Lemma 5,
(I/2) int_0^T p_v(r) dv = E H(Law(L_T | U_r) | gamma)
= E H(Law(U_r | L_T) | mu_{s+r}) + d(T) - H_r
<= (I/2) theta(r)(1/lambda + T) + d(T).
The first equality is from Lemma 5, applied conditionally on U_r.
Fix v >= 0 and then T > v. Since p_t(r) is nondecreasing in signal
strength t, Lemma 15 gives
(T - v) limsup p_v(r) <= e^{-r}(1 + T).
Taking the sequence limit with T fixed and then T arbitrarily large,
limsup p_v(r) <= e^{-r}. (7.21)
Lemma 16 applies to p = eta and q = mu_s; the exponential-square
moment was proved in Lemma 15. Since H_z <= theta(z) H_0, it gives
int_0^z m(r) dr >= (1 - theta(z))/lambda, 0 < z < 1/4.
The reverse limiting inequality follows from (7.21), 0 <= m <= 1, and
reverse Fatou. Hence
int_0^z m(r) dr -> 1 - e^{-z}. (7.22)
The function m is nonincreasing in r. For 0 < h < r with r + h < 1/4,
its value at r lies between its averages on [r, r+h] and [r-h, r].
Equation (7.22) determines the limits of both averages; h -> 0 gives
(7.20). Finally m(r) <= p_T(r) together with (7.21) proves the
strict-branch assertion for every fixed T.
Gap branch, without additional probes. Linearize Lemma 16. Let q be a
bounded smooth centered test under mu_s with bounded gradient, and put
d eta_eps = (1 + eps q) d mu_s with |eps| ||q||_infty < 1. This
alternative has finite exponential-square moment. Write
qbar_r = E_{mu_s}[q(Y_0) | U_r] and
Dqbar_r = E_{mu_s}[D q(Y_0) | U_r]. The output density ratio is
1 + eps qbar_r, and the predicted initial relative score is exactly
E_{eta_eps}[D log(1 + eps q)(Y_0) | U_r]
= eps Dqbar_r / (1 + eps qbar_r).
Its squared norm averaged under the alternative output is therefore
eps^2 E_{mu_{s+r}}[|Dqbar_r|^2 / (1 + eps qbar_r)]. The entropy
expansion H((1 + eps q) mu_s | mu_s) = eps^2 ||q||_2^2 / 2 + o(eps^2),
together with its output version, gives
||q||_2^2 - ||qbar_z||_2^2 <= int_0^z ||D qbar_r||_2^2 dr. (7.23)
The denominator is bounded away from zero uniformly in r, and
conditional Jensen bounds the integrand by a constant times ||Dq||_2^2,
justifying the limit. Approximation in the form norm extends (7.23) to
q = varphi: centered compact smooth approximants are bounded, and
conditional expectation is L^2-contractive. To retain compact support
while centering, subtract the approximant's mean times one fixed
compact smooth function of mean one; that correction tends to zero in
the form norm.
Heat-variance identity used here. For a centered scalar or
finite-dimensional vector field g in L^2(mu_s), let
g_r = E[g(Y_0) | U_r] under the base law. Then
(d/dr) E_{mu_{s+r}} |g_r|^2 = - E_{mu_{s+r}} |D g_r|^2. (7.24)
For bounded g, both the numerator density g_r mu_{s+r} and the
denominator density mu_{s+r} solve the heat equation, so the quotient
calculation with the square function gives
(d/dr - Delta/2)(mu_{s+r} |g_r|^2) = - mu_{s+r} |D g_r|^2.
Integrate against expanding cutoffs; their Laplacians vanish uniformly.
The bound |g_r| <= ||g||_infty removes the cutoff terms and gives the
integrated identity on positive time. The initial endpoint follows from
||g_r(U_r) - g(Y_0)||_2 -> 0: for bounded continuous g use g(U_r) as a
predictor and dominated convergence under the coupling
U_r = Y_0 + sqrt(r) G, then approximate in L^2(mu_s) and use
conditional contraction. Approximate a general L^2 field by bounded
ones. Applying the identity to their differences shows that the
gradients are Cauchy in the space-time energy norm, while conditional
contraction gives convergence of both endpoint norms. This proves the
integrated version of (7.24) in full generality.
Poincare at mu_{s+r} and preservation of the mean therefore give
E_{mu_{s+r}} |g_r|^2 <= theta(r) E_{mu_s} |g|^2.
Apply this first to varphi and then to w - E w. The second and (7.1)
give m(r) <= theta(r) + o(1). The first and ||varphi||_2^2 = I/lambda,
inserted into (7.23), give
int_0^z m(r) dr >= (1 - theta(z))/lambda.
The same squeeze and monotonicity argument prove (7.20). Multiplying
varphi by positive scalars multiplies every squared quantity by the
square, so their choice arranges I -> 0. QED.
7.4 Posterior covariance and outlier subspaces.
The hierarchy construction depends on two further consequences of the
prediction curve. The strict branch needs conditional covariance tails;
both branches need bounds on the subspaces where w has nonnegligible
energy.
Lemma 18. Use the critical sequence and notation of Proposition 17,
with I -> 0 in the gap branch. Then:
(i) For every fixed c < infinity,
sup over deterministic orthogonal projections Pi, rank Pi <= c I:
E |Pi w|^2 = o(I). (7.25)
(ii) For every epsilon > 0 there is a deterministic subspace
V_epsilon with dim V_epsilon <= C_epsilon I such that, for every
v perpendicular to V_epsilon,
v^*(E w w^*) v <= epsilon |v|^2,
v^*(E b_s b_s^*) v <= epsilon |v|^2,
v^*(E M_s) v <= epsilon |v|^2. (7.26)
In the gap branch V_epsilon = {0} suffices, and
||E b_s b_s^*||_op + ||E M_s||_op = o(1).
(iii) Strict branch only: for fixed r in (0,1/4) and epsilon > 0,
E tr(C_r 1_{C_r > epsilon Id}) = o(I), (7.27)
where C_r = Cov(w(Y_0) | U_r) and the indicator denotes the spectral
projection.
Proof. Strict-case spectral estimates. If S has covariance C and mean
a, the affine predictor a + C (Id + C)^{-1} (S + G - a) of S from S + G
has mean squared norm |a|^2 + tr(C^2 (Id + C)^{-1}), as a direct
expansion of the covariances shows. Conditional expectation achieves
at least this squared norm, being the orthogonal projection onto all
functions of the observation. The same holds conditionally on any side
information, with fresh Gaussian noise.
The Gaussian-signal derivative formula and monotonicity of the
prediction energy (Lemma 5) give
E |E[w | L_1]|^2 <= 2 (d(2) - d(1)) <= 2 d(2) = o(I).
Apply the affine-prediction bound to S = w and C = Cov_eta w. For every
fixed epsilon > 0,
tr(C 1_{C > epsilon Id}) = o(I), (7.28)
because t^2/(1 + t) >= epsilon t / (1 + epsilon) for t > epsilon. For
any projection of rank at most c I,
E |Pi w|^2 <= epsilon c I + tr(C 1_{C > epsilon Id}) + |E w|^2.
First sequence limit, then epsilon -> 0. Equation (7.1) proves (7.25)
uniformly in Pi.
In the gap branch, a projection of rank at most c I is eventually
zero, which proves (7.25).
Conditionally on U_r, the additional observation L_1 has signal w and
covariance C_r. Orthogonality of successive conditional expectations
and the conditional affine-prediction bound give
I (p_1(r) - m(r)) >= E tr(C_r^2 (Id + C_r)^{-1}).
Proposition 17 makes the left side o(I). The scalar inequality
t^2/(1+t) >= epsilon t/(1+epsilon) again gives (7.27).
Bounding the dimensions of the outlier spaces. Let
D = E_eta E[XX^* | X + sqrt(s) G = Y_0],
where the inner posterior kernel refers to the original prior mu.
Conditional Jensen and the definition of M_s give
E_eta b_s b_s^* <= D, E_eta M_s <= D/s. (7.29)
For each deterministic projection Q of rank l >= 1, the subgaussian
assumption of Lemma 6 gives
log E_mu exp(|QX|^2 / (C a^2)) <= C l. (7.30)
Indeed choose a 1/2-net of the unit sphere in ran Q with at most 5^l
points. |QX| is at most twice the largest absolute linear form over
this net, and bounding the exponential of a maximum by the sum proves
(7.30).
Strict branch. Form the joint law by first drawing Y_0 from eta and
then X from the original posterior kernel. Its entropy relative to the
original joint law of (X, X + sqrt(s) G) equals H_0 = I/(2 lambda).
Data processing bounds the entropy of its X-marginal relative to mu by
H_0. The entropy variational inequality applied to (7.30) gives
tr(QD) <= C a^2 (H_0 + l). (7.31)
If the spectral subspace of D above tau > 0 has dimension l, then
tau l <= C a^2 (H_0 + l). For small a this forces
l <= 2 C a^2 H_0 / tau = O_tau(I); the assertion is trivial when
l = 0. The spectral subspace of E w w^* above epsilon has dimension at
most I/epsilon, since its trace is I. Take V_epsilon to be the span of
these two spectral subspaces. Its dimension is at most C_epsilon I, and
(7.29) gives the three bounds in (7.26).
Gap branch. Averaging the original posterior under eta = mu_s gives
D = E_mu XX^* <= C a^2 Id. Equation (7.29) therefore gives the two
operator bounds directly. Also ||E w w^*||_op <= I -> 0, so
V_epsilon = {0} suffices. QED.
8. A SYMMETRIC HIERARCHY OF NEARLY FIRST EIGENFUNCTIONS
The prediction curve measures much more than the size of a conditional
mean: its Bessel inequality is asymptotically saturated at every fixed
derivative order. That saturation yields symmetric tensor fields T_l
whose derivatives remain nearly first eigenfunctions, and every tensor
has negligible energy in the curvature directions of mu_s. Throughout,
tensor norms are unnormalised Euclidean coefficient norms, including
for high order.
8.1 The hierarchy.
Fix the critical sequence of Proposition 14, retaining lambda, A, eta,
w, b_s, M_s, K_s, I. Let U_r = Y_0 + sqrt(r) G for Y_0 ~ eta and G
standard Gaussian independent of Y_0. Write
h_r(y) = E[w(Y_0) | U_r = y], h_0 = w,
m(r) = ||h_r||_2^2 / I.
Then m: [0, infty) -> [0, 1] is nonincreasing, and by Proposition 17
lim m(r) = e^{-r} for every fixed r in (0, 1/4). (8.1)
Let g_r(y) = E_G h_r(y + sqrt(r) G) be the doubly smoothed prediction.
Proposition 19 (Symmetric hierarchy). There exist symmetric tensor
fields T_l, l = 0, 1, 2, ..., of order l+1, with T_0 = w and
||T_l||_2^2 = (1 + o(1)) I, (8.2)
||D^j T_l||_2^2 = (1 + o(1)) I for every fixed j, (8.3)
T_{l+1} = D T_l + o_{L^2}(sqrt I), (8.4)
(A - 1) T_l = o_{L^2}(sqrt I), (8.5)
<T_l, K_s T_l>_2 = o(I). (8.6)
T_l is symmetric in all l+1 slots, and (8.5)-(8.6) hold also with every
one-slot flattening. Moreover, for every fixed c < infinity and every
deterministic orthogonal projection Pi acting on an existing slot,
with rank Pi <= c I,
||Pi T_l||_2^2 = o(I). (8.7)
The o(.) errors are uniform over fixed l and over the finite lists used
below.
8.2 Bessel inequality for the prediction.
The proof begins with a Gaussian-chaos decomposition of the doubly
smoothed prediction.
Lemma 20. With the data of Proposition 19,
<w, g_r>_2 = m(r) I,
sum_{j >= 0} (r^j / j!) ||D^j g_r||_2^2 <= m(r) I. (8.8)
In particular every fixed derivative in (8.8) is in the form domain
of A.
Proof. The first identity is conditional expectation:
E <w(Y), h_r(U_r)> = E |h_r(U_r)|^2.
Fix y. Gaussian integration by parts identifies the order-j Hermite
coefficient of h_r(y + sqrt(r) G) as r^{j/2} D^j g_r(y). Its projection
onto that chaos is
(r^{j/2} / j!) D^j g_r(y) : H_j(G),
where the vector output slot is not contracted. By the Hermite isometry,
the squared norm of this projection equals r^j |D^j g_r(y)|^2 / j!.
Bessel's inequality, integrated in y ~ eta, gives (8.8). QED.
8.3 Cross-covariance and spectral smoothing.
Two elementary Hilbert-space facts prevent dimension factors and
unjustified differentiations in the induction.
Lemma 21. (a) Let F be a square-integrable random variable with values
in a finite-dimensional Hilbert space H, and let Z be a centered
Euclidean random vector with Cov Z <= c Id. Then
|E(F ⊗ Z)|_HS^2 <= c E|F|^2. (8.9)
(b) Let (A^{(k)}) be a sequence of nonnegative Dirichlet operators on
fields with kernel the constants, whose spectral gap is at least
lambda -> 1. If fields S lie in the form domain of A and satisfy
|E S|^2 = o(I), ||S||_2^2 = (1 + o(1)) I,
||DS||_2^2 <= (1 + o(1)) I, (8.10)
then S~ = e^{1 - A} S satisfies
||S~ - S||_2 + ||D(S~ - S)||_2 = o(sqrt I),
|| (A - 1) S~ ||_2 = o(sqrt I). (8.11)
Proof. (a) The map u -> u.Z from H to scalar L^2(Prob) has operator
norm at most sqrt(c). Its adjoint sends a scalar coefficient to
E[Z u], and summing across coefficients gives
|E(F ⊗ Z)|_HS^2 = sum_k |E(F_k Z)|^2
<= c sum_k E|F_k|^2 = c E|F|^2.
(b) Let E_S(mu) denote the spectral resolution of A applied to S. The
identities
||S||_2^2 = int_0^infty d||E_S(mu) S||_2^2,
||DS||_2^2 = int_0^infty mu d||E_S(mu) S||_2^2,
together with (8.10), give
int_{mu >= lambda} (mu - lambda) d||E_S S||_2^2
<= ||DS||_2^2 - lambda ||S||_2^2 + lambda |E S|^2
= o(I). (8.12)
Spectral mass at mu = 0 carries weight |E S|^2 = o(I), and (8.12)
bounds the mass of the remaining contribution at mu >= 1 + delta by
o(I)/delta for every fixed delta > 0. Splitting the multiplier
e^{1-mu} - 1 into the ranges {mu = 0}, [lambda, 1 + delta],
(1 + delta, K], (K, infty) for large K: the first and fourth give
o(I), the second is O(delta) by |e^{1-mu} - 1| <= C |mu - 1| near
mu = 1, and the third is at most sqrt(o(I)/delta). Sending delta -> 0
after the sequence limit, with K free, proves ||S~ - S||_2 = o(sqrt I).
For the derivative norm,
||D(S~ - S)||_2^2 = int mu |e^{1-mu} - 1|^2 d||E_S S||_2^2.
On [lambda, 1 + delta] the factor mu |e^{1-mu} - 1|^2 is at most
C delta^2. On (1 + delta, K] it is at most K, and the mass there is
bounded by o(I)/delta. On (K, infty) it is at most mu, and
int_{mu > K} mu d||E_S S||_2^2 = o(I) by (8.12) with lambda -> 1.
This proves ||D(S~ - S)||_2 = o(sqrt I).
The spectral multiplier of (A - 1) S~ is (mu - 1) e^{1 - mu}, vanishing
at mu = 1 and bounded on [0, infty). The same piecewise splitting
gives
int |(mu - 1) e^{1-mu}|^2 d||E_S S||_2^2
<= C delta^2 ||S||_2^2 + o(I)/delta = O(delta^2) I + o(I)/delta.
Sending delta -> 0 after the sequence limit gives (8.11). QED.
8.4 Hierarchy construction.
The construction starts at T_0 = w and produces T_{l+1} from T_l. Two
ingredients combine: Bessel tightness (Lemma 20) supplies an
L^2-close symmetrized candidate, and the smoothing estimate of Lemma 21
converts that candidate into a nearly first eigenfunction. The
curvature estimate (8.6) requires separate machinery, deferred to
Section 8.7.
Lemma 22 (Construction induction). Suppose T_0,...,T_l satisfy
(8.2)-(8.5) and (8.7) at orders 0,...,l. Then there exists a symmetric
field T_{l+1} of order l+2 satisfying (8.2)-(8.5) and (8.7) at order
l+1, with E T_{l+1} = 0. The starting vector obeys |E w|^2 <= o(I)
by (7.1).
Proof. Set V = DT_l and L = l+1.
Step 1. Self-adjointness and (8.4) give
||V||_2^2 = <T_l, A T_l> = ||T_l||_2^2 + o(I) = (1 + o(1)) I.
For any admissible projection Pi on the derivative slot,
||Pi_1 V||_2^2 = <Pi_1 T_l, A Pi_1 T_l>
<= ||Pi_1 T_l||_2^2 + ||Pi_1 T_l||_2 ||(A-1) T_l||_2 = o(I),
uniformly in Pi, by (8.7) and (8.5).
Step 2. Fix 0 < r < 1/4. Put V_j = T_j for 0 <= j <= l, and V_L = V.
Successive form integration by parts gives
|I^{-1} <D^j g_r, V_j>_2 - e^{-r}| = o(1), 0 <= j <= L. (8.22)
At j = 0 this is (8.8) with m(r) -> e^{-r}. At 1 <= j <= L,
<D^j g_r, DT_{j-1}>_2 = <D^{j-1} g_r, A T_{j-1}>_2
= <D^{j-1} g_r, T_{j-1}>_2 + o(I),
by ||(A-1)T_{j-1}||_2 = o(sqrt I) and (8.8). At j = L the same
identity applies with T_{l} = T_{L-1} and DT_l = V.
Step 3. Multiply (8.8) by e^{2r}/I and use step 2 with
||V_j||_2^2 = (1+o)I. Put
kappa_r(L) = e^r - sum_{j=0}^L r^j/j! > 0. (8.23)
The exponential-series tail absorbs everything beyond the j <= L
contributions matched in step 2:
limsup ||e^r D^L g_r - V||_2^2 / I
<= (L! / r^L) kappa_r(L) = O_L(r), (8.24)
limsup e^{2r} ||D^{L+1} g_r||_2^2 / I
<= ((L+1)! / r^{L+1}) kappa_r(L) = 1 + O_L(r). (8.25)
Step 4. Choose r -> 0 slowly along the sequence so that
S_0 := e^r D^L g_r satisfies
||S_0 - V||_2 = o(sqrt I), ||DS_0||_2^2 <= (1 + o(1)) I. (8.26)
S_0 is symmetric in its last L slots, V is symmetric in the first L
since V = DT_l and T_l is symmetric. For L >= 2 the adjacent
transpositions within each block generate the full symmetric group
S_{L+1}; for L = 1, V = Dw is already symmetric because w = Dh. Let
S be the full symmetrization of S_0. Then
||S - V||_2 = o(sqrt I), ||S||_2^2 = (1 + o(1)) I,
||DS||_2^2 <= (1 + o(1)) I. (8.27)
Averaging coefficient permutations is an orthogonal contraction
commuting with differentiation, and step 1 transfers to every
existing slot of S.
Step 5. Write C_y = Cov_eta(T_l, Y), C_b = Cov_eta(T_l, b_s),
C_w = Cov_eta(T_l, w). Testing A T_l = T_l + (A-1)T_l against
Y - E Y and using self-adjointness with the coordinate-generator
A_eta Y = (Y - b_s)/s - w in the form domain gives
E V = C_y + E_l, |E_l|_HS <= lambda^{-1/2} ||(A-1)T_l||_2, (8.28)
E V = s^{-1} (C_y - C_b) - C_w. (8.29)
The error bound uses cross-covariance (8.9) with Cov_eta Y <=
lambda^{-1} Id (Poincare). The bounds E_eta b_s b_s^* <= C a^2 Id and
|E_eta w|^2 = o(I) of Lemma 12 combine with (8.28)-(8.29):
|E V|_HS = o(sqrt I), |C_y|_HS = o(sqrt I). (8.30)
Pass to the centered field S_c = S - E S. Then
||S_c - V||_2 = o(sqrt I), |E S_c| = 0,
||DS_c||_2 = ||DS||_2 <= (1 + o(1)) I,
||S_c||_2^2 = (1 + o(1)) I. (8.31)
Step 6. Apply Lemma 21(b) to S_c; the spectral gap of A_eta is at
least lambda -> 1. The field T_{l+1} := e^{1-A} S_c satisfies
||T_{l+1} - S_c||_2 + ||D(T_{l+1} - S_c)||_2 = o(sqrt I),
||(A - 1) T_{l+1}||_2 = o(sqrt I). (8.32)
Combining (8.31) and (8.32):
||T_{l+1} - V||_2 = o(sqrt I),
||T_{l+1}||_2^2 = (1 + o(1)) I,
||D T_{l+1}||_2^2 <= (1 + o(1)) I,
||(A - 1) T_{l+1}||_2 = o(sqrt I).
Since T_{l+1} - V = o(sqrt I) and V = DT_l, (8.4) holds at order
l+1. (8.2) and (8.3) at every fixed derivative order hold by
construction. Symmetry passes into T_{l+1} because e^{1-A} commutes
with coefficient permutations. (8.7) transfers from S to T_{l+1} by
step 4 and (8.32).
Step 7. The field T_{l+1} is centered: E S_c = 0 and semigroups
preserve the mean (1 is in the kernel of A), so
E T_{l+1} = e E S_c = 0. This completes the induction for (8.2)-(8.5)
and (8.7). The curvature assertion (8.6) is deferred to Section 8.7.
QED.
8.5 Vanishing operator tails.
The curvature estimate of Section 8.7 tests the operator identity
D(A F) = A(D F) + (K_s - Q) D F against a field P whose operator norm
is small, so that Q P is genuinely negligible in L^2. That is what the
next lemma supplies.
Lemma 23 (Vanishing operator tails). In the strict branch put Q = Dw;
in the gap branch Q = 0. For every fixed delta > 0 and every l >= 1,
each one-slot flattening F of T_l, and also F = Q in the strict branch,
obeys
E tr(F F^* 1_{F F^* > delta^2 I}) = o(I). (8.33)
Equivalently, clipping every singular value at any fixed positive
cutoff changes the field by o(sqrt I) in L^2.
Proof. Base case. Take F = Q or F = T_1. Lemma 22 at L = 1 gives
||e^r D g_r - Q||_2^2 <= (r/2 + O(r^2)) I, (8.34)
so Q = Dw is L^2-close to a Gaussian-averaged derivative of a smooth
field. A diagonal-smoothing approximation replaces T_1 by the same
field up to o(sqrt I). The conditional covariance bound of Lemma 18(iii)
puts the singular values of the local posterior covariances of w below
any fixed threshold outside a subspace of vanishing trace;
Gaussian-convolution regularity propagates this into pointwise
operator-norm bounds on D g_r over compact y-sets, and the
exponential-square moment of eta makes the outside mass vanish.
Splitting ||Q 1_{||Q||_op > delta}||_HS^2 into {|y| <= R} and {|y| > R}:
||Q 1_{||Q||_op > delta}||_HS^2
<= 2 ||D g_r 1_{||D g_r||_op > delta/2}||_2^2 + o(I) = o(I), (8.35)
after sending r -> 0 slowly along the sequence, then R -> infty, then
the sequence limit. The bound transfers to T_1 through
||T_1 - Q||_2 = o(sqrt I).
Propagation to l >= 2. Fix l >= 1 and t > 0. Apply the Hermite
expansion to the stationary diffusion at time t, with starting level 1.
Conditional on Y_0, the Gaussian-Hermite chaoses of P_t T_1 are
orthogonal even after acting on the first tensor slot by an arbitrary
deterministic orthogonal projection P(Y_0) (a function of Y_0 alone).
In the expansion of P_t T_1 against the diffusion, the chaos containing
T_l(Y_0) carries squared coefficient
c_{l,t} = e^{-2t} (2t)^{l-1} / (l-1)! > 0. (8.36)
The eigenvalue exp(-t) of A contributes the exp(-2t) factor, the
l-1 derivatives from T_l = (D^{l-1} T_1) contribute (2t)^{l-1},
and projection is norm one on each chaos. Take the expansion to any
fixed order m >= l-1, take the sequence limit at fixed m, then let
m -> infty in the uniform remainder bound. The result is
liminf((E|P(Y_0)_1 T_1(Y_t)|^2
- c_{l,t} E|P(Y_0)_1 T_l(Y_0)|^2) / I) >= 0. (8.37)
The remainder estimates are uniform in P, whose operator norm is at
most one.
Conditional contraction of P_t against P(Y_0) gives
E|P(Y_0)_1 P_t T_1(Y_0)|^2 <= E|P(Y_0)_1 T_1(Y_t)|^2.
Combining with (8.37),
c_{l,t} E|P(Y_0)_1 T_l(Y_0)|^2
<= E|P(Y_0)_1 T_1(Y_t)|^2 + o(I). (8.38)
Choose P(Y_0) as the spectral projection of the first-slot flattening
of T_l(Y_0) above threshold epsilon. Then
E rank P <= epsilon^{-2} ||T_l||_2^2
= (1 + o(1)) epsilon^{-2} I. (8.39)
Let S_delta be the singular-value clipping of T_1 at delta. Pointwise
|P(Y_0) S_delta(Y_t)|_HS^2 <= delta^2 rank P(Y_0). (8.40)
Stationarity and the base case give
||S_delta(Y_t) - T_1(Y_t)||_2 = o(sqrt I) uniformly in t. Hence
E|P(Y_0)_1 T_1(Y_t)|^2
<= delta^2 (1+o) epsilon^{-2} I + o(I). (8.41)
Combine (8.38) and (8.41):
c_{l,t} E|P(Y_0)_1 T_l(Y_0)|^2
<= delta^2 epsilon^{-2} I + o(I). (8.42)
Send delta -> 0. The left side equals
E tr(T_l T_l^* 1_{T_l T_l^* > epsilon^2 I}) on the first-slot
flattening. This proves (8.33) at threshold epsilon for the
first-slot flattening of T_l. Tensor symmetry supplies every other
one-slot flattening, and exchanging the roles of P(Y_0) and T_1(Y_t)
preserves the bound, so no independence between P(Y_0) and T_1(Y_t)
was needed. QED.
8.6 Weighted divergence and the differentiated operator.
The next identity is the only place where differentiating the drift is
legitimate without uniform control of third derivatives of the weighted
density. It pairs the weighted divergence of a matrix field against a
function in the operator domain, using only the second-order potential.
Lemma 24. Let eta = e^{-V} be a smooth strictly positive probability
density with A_eta = -Delta - DV.D and H = D^2 V. Let P = (P_{i alpha})
have one vector slot i and a finite-dimensional coefficient index alpha.
Suppose every coefficient lies in Dom(A_eta) and H P lies in L^2(eta).
Then the weighted adjoint divergence
(D^* P)_alpha = - sum_i d_i P_{i alpha}
- sum_i (d_i V) P_{i alpha} (8.47)
belongs to L^2(eta) and satisfies
||D^* P||_2^2 <= ||DP||_2^2 + <P, H P>_2. (8.48)
For every coefficient field F in Dom(A_eta),
<AF, D^* P>_2 = <DF, AP>_2 + <DF, H P>_2. (8.49)
In particular (8.49) requires H P in L^2 and does not require H DF
in L^2.
Proof. For compactly supported smooth P, two first derivatives commute
in weighted integration by parts, so
||D^* P||_2^2
= sum_{i, j, alpha} <d_j P_{i alpha}, d_i P_{j alpha}>_2
+ <P, H P>_2. (8.50)
Cauchy-Schwarz applied to the arrays (i, j, alpha) and (j, i, alpha)
bounds the double sum by ||DP||_2^2 without any factor from the numbers
of coordinate or coefficient indices. This proves (8.48) for smooth
compactly supported P.
For the stated P, membership in Dom(A_eta) puts P in H^2_loc by local
elliptic regularity, and D P in L^2(eta). Choose smooth compactly
supported cutoffs 0 <= chi_R <= 1 with chi_R -> 1 monotonically and
||D chi_R||_infty -> 0. Equation (8.50) applies to chi_R P by local
Sobolev approximation, and its right side is controlled by
||D(chi_R P) - DP||_2 -> 0,
<chi_R P, H chi_R P>_2 -> <P, HP>_2.
The second convergence uses |<P, H P>| in L^1, a consequence of P, HP in
L^2. Weak compactness in L^2 and distributional convergence of
D^*(chi_R P) identify the limit as D^* P. Lower semicontinuity yields
(8.48). Only first derivatives of the cutoffs enter the estimate.
For compactly supported smooth F, differentiating the scalar operator
D(A F) = A(D F) + H D F. (8.51)
Pairing this identity with P and using symmetry of H gives (8.49) for
such F. For general F in Dom(A_eta), take compactly supported smooth
F_j with F_j -> F and A F_j -> AF in L^2, as Lemma 3 supplies. The same
convergence holds in the form norm, since
||D(F_j - F)||_2^2 = <F_j - F, A(F_j - F)>_2 -> 0.
All three factors D^* P, AP, HP in (8.49) lie in L^2, so every term
converges. This proves the global identity. QED.
8.7 The curvature estimate: completion of Proposition 19.
Lemma 22 supplies the hierarchy fields and every assertion of
Proposition 19 except (8.6). What remains is to show
<T_l, K_s T_l>_2 = o(I) at each level, using Lemma 24 and the operator
tails of Lemma 23.
Step 1: The test field P. Fix l >= 0 and write F = T_l. View DF as a
matrix with its derivative slot as row index and all old tensor slots
as column index.
Strict branch. Apply Lemma 23 with a fixed delta > 0 to the
corresponding flattening of T_{l+1}. This gives a clipped field C with
||C - T_{l+1}||_2 = o(sqrt I),
||C(y)||_op <= delta. (8.66)
Set P = e^{1-A} C = e P_1 C. Markov averaging and convexity of the
operator-norm ball give ||P(y)||_op,infty <= e delta. The spectral
multipliers (x-1) e^{1-x}, e^{1-x}, x^{1/2} e^{1-x} are bounded on
[0, infty); the function e^{1-x} - 1 vanishes at x = 1 and satisfies
|e^{1-x} - 1| <= C |x - 1| there. Combining these bounds with (8.5)
and the derivative-closeness T_{l+1} = DF + o(sqrt I):
||P - DF||_2 + ||(A-1) P||_2 = o(sqrt I), (8.67)
||DP||_2 = O(sqrt I), (8.68)
||QP||_2 <= ||P||_op,infty ||Q||_2 <= e delta (1+o(1)) sqrt I. (8.69)
The clipping is used only to construct P, so it need not preserve
tensor symmetry.
Gap branch. Take P = T_{l+1} directly. Then Q = 0 and (8.67)-(8.69)
follow from (8.3), (8.4), and (8.5).
In both branches the field P obeys (8.67)-(8.69), P lies in Dom(A), and
H P lies in L^2(eta) for H = K_s - Q.
Step 2: The curvature identity. Lemma 22 gives the drift decomposition
H = D^2(-log eta) = K_s - Q, (8.70)
with Q = 0 in the gap branch. Both K_s P and Q P lie in L^2(eta), since
||K_s P||_2 <= s^{-1}||P||_2 and ||QP||_2 = o(sqrt I). Lemma 24 applies
to F = T_l and P:
<A F, D^* P>_2 = <DF, AP>_2 + <DF, (K_s - Q)P>_2. (8.71)
Insert A F = F + (A-1)F and A P = P + (A-1)P. The divergence identity
for weighted integration by parts,
<F, D^* P>_2 = <DF, P>_2, (8.72)
cancels the two zeroth-order terms. Therefore
<DF, K_s P>_2
= <(A-1)F, D^* P>_2 - <DF, (A-1)P>_2
+ <DF, QP>_2. (8.73)
Bounds:
- ||D^* P||_2^2 <= ||DP||_2^2 + <P, H P>_2 <= O(I) + s^{-1}||P||_2^2
+ ||P||_2 ||QP||_2 = O(I). So |<(A-1)F, D^* P>| <= ||(A-1)F||_2
||D^* P||_2 = o(sqrt I) O(sqrt I) = o(I).
- |<DF, (A-1)P>| <= ||DF||_2 ||(A-1)P||_2 = O(sqrt I) o(sqrt I) = o(I).
- |<DF, QP>| <= ||DF||_2 ||QP||_2 = O(sqrt I) e delta O(sqrt I) = o(I)
as delta -> 0.
Together,
<DF, K_s P>_2 = o(I). (8.74)
Replace P by DF. Since P - DF = o(sqrt I) and ||K_s||_op,infty <= s^{-1},
|<DF, K_s (P - DF)>| <= ||DF||_2 s^{-1} ||P - DF||_2 = o(I). (8.75)
So <DF, K_s DF>_2 = o(I). Replacing DF by T_{l+1} costs o(I) by the
derivative estimate: <T_{l+1}, K_s T_{l+1}>_2 = o(I).
Step 3: Every slot. The identity <T_{l+1}, K_s T_{l+1}>_2 = o(I) now
holds on the derivative slot. By tensor symmetry (8.6) holds on every
slot. Level zero is the initial vector: E_eta <w, K_s w> <= lambda
|E_eta w|_2^2 = o(I) by Lemma 12 and |E w|^2 = o(I) from (7.1). Every
assertion of Proposition 19 is in place. QED.
9. A MATRIX FIELD AND A FROZEN DIFFUSION LAW
The hierarchy has fixed finite levels, but no single well-defined matrix.
Packing a band of levels produces a positive matrix field with sharp
energy, from which a Gaussian matrix fluctuation law emerges. The
diagonal choice made at the start of this section fixes every finite
auxiliary constant used below.
9.1 The band reduction.
Diagonal choice. For each finite integer K, impose the conclusions of
Proposition 19 through level 3K, the fixed-projection inequalities for
rank coefficients at most K, and the Hermite expansion bounds for every
starting level 0 <= l < 2K and every truncation order 0 <= m <= K, on
time intervals 0 <= t <= K. Only levels through 3K enter these
expansions, including their remainders. For every finite list, the
hierarchy defects and the excesses over the normalized Hermite
remainder bounds can be made arbitrarily small. Choose a positive
integer k that increases along the sequence slowly enough that all these
defects vanish uniformly. Further parameters below are chosen by the
same procedure. Normalized endpoint energies remain bounded when a band
of k levels is shifted by a fixed number of levels.
Regard
Psi = direct_sum_{k <= l < 2k} T_l / sqrt(k I) (9.1)
as a matrix by separating its first tensor index from the remaining
indices and the direct-sum label. Define
rho = Psi Psi^*, R = rho^{1/2}. (9.2)
The matrix field R is positive semidefinite; it is distinct from the
scalar log-Sobolev constant R(eta), and context distinguishes them. The
normalization reads
E tr R^2 = ||Psi||_2^2 -> 1,
||D Psi||_2^2 -> 1. (9.3)
9.2 Curvature and centering.
Two consequences of the hierarchy pass to the matrix field.
First, the curvature conclusion (8.6), at every level 0 <= l < 2k,
<T_l, K_s T_l>_2 = o(I).
Summing over the band with weights 1/(k I) identifies the left side
with tr(K_s rho) up to a uniform o(1). Since K_s = (Id - M_s)/s,
E tr((1 - M_s) R^2) = o(1). (9.4)
Second, an independence statement. Fix delta > 0 and let P_delta be the
spectral projection of E M_s above delta. Lemma 18(ii) supplies
rank P_delta = O_delta(I), and the deterministic-projection bound (8.7)
gives
tr(P_delta E R^2) = o(1). (9.5)
Since 0 <= M_s <= Id, this bounds the left side of
tr((E M_s)(E R^2)) by o(1) + delta E tr R^2 = o(1) + delta. Send
delta -> 0 after the sequence limit:
tr((E M_s)(E R^2)) = o(1). (9.6)
In the gap branch of Lemma 18, ||E M_s||_op = o(1) makes (9.6) immediate.
9.3 Energy lower bound for R.
Let R' denote R evaluated at an independent copy of Y_0. Insert
(1 - M_s) + M_s between R and R' in ||E R||_HS^2 = E tr(R R').
Hilbert-Schmidt Cauchy-Schwarz bounds the term containing 1 - M_s by
(E tr((1 - M_s) R^2))^{1/2} (E tr R^2)^{1/2} = o(1), (9.7)
using (9.4) and (9.3).
The remaining term is bounded by
(E tr(M_s R^2))^{1/2} (tr((E M_s)(E R^2)))^{1/2} = o(1), (9.8)
because E tr(M_s R^2) <= E tr(R^2) is at most 1 + o(1) and
tr((E M_s)(E R^2)) = o(1) by (9.6).
Together,
||E R||_HS = o(1). (9.9)
Poincare for eta and (9.3) give
Var_eta(R) = E tr R^2 - ||E R||_HS^2 = 1 + o(1),
Var_eta(R) <= R_eta^P ||D R||_2^2 <= (1 + o(1)) ||D R||_2^2,
so
||D R||_2^2 >= 1 + o(1). (9.10)
The upper bound is the elementary inequality
||D R||_2^2 <= ||D Psi||_2^2 = 1 + o(1), following from the chain rule
D rho = D R · R + R · D R applied to the positive square root and the
normalized energy of Psi in (9.3). Consequently
||D R||_2^2 = 1 + o(1). (9.11)
9.4 Averaging over symmetric tensor slots.
The next estimate is the reason for using tensors with many symmetric
slots. It compares averaging an action over those slots with the
geometric, rather than arithmetic, weighting by the marginal matrix.
Lemma 25. Let psi be a finite direct sum of symmetric tensors, each
with at least k >= 2 slots. Let rho be its one-slot reduction and
R = rho^{1/2}. For a self-adjoint matrix H let H_av act on each
component by averaging the action of H over its slots. Then
||H_av psi||_2^2 <= tr(R H R H) + (2/k) tr(rho H^2). (9.12)
Proof. For a component psi_d with d slots put
a_d = <psi_d, H_(1)^2 psi_d>_2,
c_d = <psi_d, H_(1) H_(2) psi_d>_2. (9.13)
Symmetry and Cauchy-Schwarz give |c_d| <= a_d, and direct expansion
gives
||H_av psi||_2^2 = sum_d (a_d/d + (1 - 1/d) c_d)
<= sum_d c_d + (2/k) sum_d a_d. (9.14)
It remains to prove |sum_d c_d| <= tr(R H R H). This correlation bound
is a case of Lieb's Wigner-Yanase trace-concavity theorem. A direct
proof by weighted interpolation runs as follows.
Define the linear map C by
tr(X C(Y)) = sum_d <psi_d, X_(1) Y_(2) psi_d>_2. (9.15)
Flattening all remaining indices exhibits C(Y) as a sum of terms
A_alpha Y^T A_alpha^*. Thus C is positive and C(1) = C^*(1) = rho.
On the support of rho, on which inverse powers are defined, put
U(Y) = rho^{-1/2} C(Y) rho^{-1/2}. (9.16)
The map U is unital and operator-norm contractive: its preceding
representation is an isometric compression of copies of Y^T. It also
satisfies tr(rho U(Y)) = tr(rho Y). For 0 <= Re z <= 1 define
U_z(Y) = rho^{z/2} U(rho^{-z/2} Y rho^{-z/2}) rho^{z/2}. (9.17)
On Re z = 0, multiplication on either side by imaginary powers of
rho is isometric, so U_z is operator-norm contractive. At z = 1,
U_1(Y) = C(rho^{-1/2} Y rho^{-1/2}), U_1^*(1) = 1. (9.18)
Its adjoint is a unital positive map with the same compression and
transpose representation, hence is operator-norm contractive. Duality
makes U_1 trace-norm contractive; imaginary powers give the same at
Re z = 1.
The Wigner-Yanase / Wigner interpolation theorem, applied on the strip
0 <= Re z <= 1 to the family U_z evaluated at Y = rho^{1/2} H rho^{1/2},
bounds |tr(U(H) rho^{1/2} H rho^{1/2})| by the endpoint norms. Since
the boundary z = 0 is operator-norm contractive and z = 1 is
trace-norm contractive, the strip argument yields
|sum_d c_d| = |tr(C(H) H)|
= |tr(U(H) rho^{1/2} H rho^{1/2})|
<= tr(R H R H). (9.19)
Substitute this in (9.14). QED.
9.5 Saturation via the Sylvester equation.
Lemma 26 (Saturation of the square-root energy). There are measurable
symmetric matrix fields H_i, 1 <= i <= n, solving the Sylvester
equation
D_i R = (H_i R + R H_i) / 2 (9.20)
with zero kernel-kernel block, such that
||D R||_2^2 -> 1, sum_i E ||[H_i, R]||_HS^2 = o(1),
sum_i E tr(R^2 H_i^2) -> 1. (9.21)
Moreover
sum_i (||D_i R - H_i R||_2^2
+ ||D_i Psi - (H_i)_(1) Psi||_2^2) = o(1). (9.22)
The first slot can be replaced by any existing slot in each component.
Proof. In an eigenbasis of R with eigenvalues r_a, define
(H_i)_{ab} = 2 (D_i R)_{ab} / (r_a + r_b) (9.23)
when r_a + r_b > 0, and zero otherwise. The kernel-kernel block of
D_i R vanishes almost everywhere. Indeed, for constant vectors v, w in
the kernel of R, the quadratic form v^* R w vanishes identically in
position; differentiating gives v^* (D_i R) w = 0. Since R is self-
adjoint and the Sylvester formula is symmetric in (a, b), the matrix H_i
is symmetric. The formula is measurable.
Energy identities. Set
e = ||D R||_2^2, a = sum_i E tr(R^2 H_i^2),
c = sum_i E ||[H_i, R]||_HS^2, d = ||D Psi||_2^2. (9.24)
From the Sylvester equation, |(H_i R)_{ab}|^2
= 4 |(D_i R)_{ab}|^2 r_b^2/(r_a + r_b)^2 <= 4 |(D_i R)_{ab}|^2,
because 0 <= r_b <= r_a + r_b. Summing over (a, b) and i,
sum_i ||H_i R||_HS^2 <= 4 ||D R||_2^2. (9.25)
Writing A = H_i R and B = R H_i, the parallel and perpendicualr parts
satisfy ||A + B||^2 - ||A - B||^2 = 4 <A, B>_HS = 4 tr(R H_i R H_i),
and ||A||_HS^2 = ||B||_HS^2 = a_i by symmetry of H_i. Therefore
e = a - c/4,
sum_i E tr(R H_i R H_i) = e - c/4. (9.26)
Differentiating R^2 = Psi Psi^* in the weak form sense and using
symmetry of D_i Psi on its slots:
e = sum_i E <D_i Psi, (H_i)_(1) Psi>_2
= sum_i E <D_i Psi, (H_i)_av Psi>_2. (9.27)
Lemma 25 gives ||(H_i)_av Psi||_2^2 <= tr(R H_i R H_i) + (2/k) tr(rho H_i^2).
Summing over i and applying Cauchy-Schwarz to (9.27):
e^2 <= d (e - c/4 + 2a/k). (9.28)
Since a <= 4e by (9.25) and d -> 1 by (9.3), equation (9.28) forces
limsup e <= 1. Combined with liminf e >= 1 from (9.10), this gives
e -> 1; then c -> 0 and a -> 1.
Final identities. From the Sylvester equation,
||D_i R - H_i R||_2^2 = (1/4) ||[H_i, R]||_HS^2, (9.29)
||(H_i)_(1) Psi||_2^2 = a_i = tr(R^2 H_i^2). (9.30)
Therefore
sum_i ||D_i R - H_i R||_2^2 = c/4 -> 0,
sum_i ||D_i Psi - (H_i)_(1) Psi||_2^2 = d + a - 2e -> 0. (9.31)
Tensor symmetry of Psi in each component lets the first slot stand
for any existing slot. QED.
9.6 Bounded columns and balance.
Lemma 27 (Balance and bounded column). The matrices H_i of Lemma 26
satisfy
E ||sum_i H_i rho H_i - rho||_1 = o(1),
E ||R (sum_i H_i^2) R - rho||_1 = o(1). (9.45)
They can be replaced by symmetric matrix fields, still denoted H_i, so
that (9.20), the commutator conclusion of (9.21), and (9.45) hold, and
S := sum_i H_i^2 <= 4 Id,
E ||(S - 1) R||_HS^2 = o(1). (9.46)
The total change satisfies
sum_i E ||(H_i^new - H_i^old) R||_HS^2 = o(1). (9.47)
Proof.
Perturbation bound. From the identity X X^* - Y Y^* = (X - Y) X^*
+ Y (X - Y)^*,
||X X^* - Y Y^*||_1
<= ( ||X||_HS + ||Y||_HS ) ||X - Y||_HS. (9.48)
Balance identities. By (9.22), the sum sum_i E ||D_i Psi - (H_i)_(1) Psi||_2^2
is o(1), so the one-slot reduction of D Psi with the derivative index
included among the column indices is close in expected trace norm to
sum_i H_i rho H_i. The hierarchy coupling identifies this reduction
with
(1 / (k I)) sum_{k <= l < 2k} T_{l+1} T_{l+1}^*. (9.49)
The trace-norm error from the coupling is o(1) by (9.48) and (9.3).
Shifting the band by a fixed number of levels changes its endpoints by
O(1/k), so the first line of (9.45) holds.
Comparing the arrays (H_i R)_i and (R H_i)_i proves the second line of
(9.45) by the same inequality, using the summed commutator estimate in
(9.21).
Bounded columns. Write H v = (H_i v)_i, S = H^* H, J = (1 + S)^{-1},
and
K = H R - R^{direct sum n} H. (9.50)
By (9.22), E ||K||_HS^2 = o(1). Algebra gives
[S, R] = H^* K - K^* H,
[R, J] = J [S, R] J. (9.51)
The bounds ||H J H^*||_op <= 1, ||H J||_op <= 1/2, ||J||_op <= 1 yield
||H [R, J]||_HS <= 2 ||K||_HS. (9.52)
For every fixed function f(x) = a + b/(1 + x) with bounded real a, b,
E ||H [f(S), R]||_HS^2 = o(1). (9.53)
The squared norms of H f(S) R and H R f(S) differ by o(1): the latter
has bounded mean-square norm, and (9.53) controls their difference.
Testing the second balance equation in (9.45) against the bounded matrix
f(S)^2 consequently gives
E tr(R^2 (S - 1) f(S)^2) = o(1). (9.54)
Inserting f = 1, f = 1 + J, f = 1 - J and the identity
(S - 1)^2/(1 + S) = (S - 1) - 2 (S - 1) J yields
E tr(R^2 ((S - 1)^2 / (1 + S))) = o(1). (9.55)
All expressions are integrable: their absolute values are bounded by a
constant multiple of tr(R^2 (1 + S)).
Tail truncation. Let Q = 1_{(4, infty)}(S), P = 1 - Q. On this spectral
region (S - 1)^2/(1 + S) dominates both a positive constant and a
positive multiple of S. Thus
E tr(R^2 Q) + E tr(R^2 S Q) = o(1). (9.56)
The input truncation costs
sum_i E ||H_i Q R||_HS^2 = E tr(R^2 S Q) = o(1).
The output truncation costs
sum_i E ||Q H_i R||_HS^2
= E tr(Q sum_i H_i rho H_i)
<= E tr(Q rho) + E ||sum_i H_i rho H_i - rho||_1 = o(1).
So replacing H_i by P H_i P has total right-R squared cost o(1). The
new column obeys sum_i (P H_i P)^2 <= P S P <= 4 P.
Perturbation accounting. Suppose symmetric Htilde_i have
Delta := sum_i E ||(Htilde_i - H_i) R||_HS^2 = o(1). (9.57)
Then symmetry gives the same bound with R on the left. Each derivative
error in (9.22) changes in norm by at most sqrt(Delta), and the summed
commutator norm changes by at most 2 sqrt(Delta). The two balance
errors change in expected trace norm by at most
( (sum_i E ||H_i R||_HS^2)^{1/2}
+ (sum_i E ||Htilde_i R||_HS^2)^{1/2} ) sqrt(Delta),
by (9.48), which is o(1). The new summed energy is bounded by the old
plus the perturbation in norm. These statements apply to the
compression just made. QED.
9.7 Spectral blocks and a bounded input map.
Ordering the operator column by the spectral range of R decouples its
action in the later Gaussian test. The matrix field R varies with
position, so its eigenbasis is not locally differentiable; a measurable
rescaling of the underlying probability space avoids addressing this
diagonalization directly. The block matrices act on the algebraic
V_beta = direct-summand of the rounded operator.
Lemma 28. After enlarging the initial probability space by an
independent uniform variable, there are a positive block matrix
R_* = direct_sum_beta beta Id_beta, with zero on the kernel of R, and
symmetric matrices H_i supported on entries whose row, column, and
outer index belong to the same positive block, satisfying:
(i) Their total change obeys
sum_i E ||(H_i^new - H_i^old) R||_HS^2 = o(1).
(ii) On the positive support, R_* <= R <= (1 + epsilon) R_* with
epsilon -> 0, and sum_i H_i^2 <= 4 Id.
(iii) With the positive weighted trace
tau(U) = E sum_beta beta^2 tr_beta U_beta, (9.61)
where the expectation also includes any fresh noises present,
tau(1) -> 1,
tau((S - 1)^2) = o(1),
sum_{i, j} tau(|[H_i, H_j]|^2) = o(1),
E sum_beta beta^2 sum_{i, a, b in beta}
|(H_i)_{ab} - (H_a)_{ib}|^2 = o(1). (9.62)
Proof. Rounding. Let
kappa = sum_i E ||[H_i, R]||_HS^2,
a = sum_i E ||H_i R||_HS^2 (9.63)
be the commutator and squared-energy functionals of Lemma 26. Both are
bounded, and kappa = o(1). Let 0 < epsilon -> 0 slowly along the
sequence, h = log(1 + epsilon), and take theta uniform on [0, h).
Group positive eigenvalues of R into ranges [e^{theta + m h},
e^{theta + (m+1) h}) for m in Z, rounding each nonempty group down to
beta = e^{theta + m h}. The zero eigenspace is kept separate. Write
R_* for the resulting block matrix.
For 0 < delta < h, a pair of positive eigenvalues with
|log(r_a / r_b)| >= delta satisfies r_b^2 <= C delta^{-2}
(r_a - r_b)^2 whenever delta <= 1. If their logarithmic distance is
smaller than delta, the probability that a block boundary separates
them is at most delta/h. Deleting the entries of H_i with row and column
in different blocks and replacing the remaining eigenvalues by their
block value gives a right-R squared cost bounded by
C delta^{-2} kappa + (delta/h) a + kappa. (9.64)
The final kappa covers a zero row and positive column directly, since
a zero column has no right-R cost. Choose delta -> 0 slowly enough
along the sequence that kappa/delta^2 -> 0, then h -> 0 slowly enough
that delta/h -> 0. This produces (i) and (ii) with sum_i H_i^2 <= 4 Id
from Lemma 27.
Weighted trace. On the block matrix R_*, tr R_*^2 = sum_beta beta^2
(rank V_beta), so the weighted trace (9.61) satisfies tau(1) = E tr R_*^2.
Because R_* <= R <= (1 + epsilon) R_*,
E tr R_*^2 <= E tr R^2 <= (1 + epsilon)^2 E tr R_*^2,
and E tr R^2 -> 1 from (9.3), so tau(1) -> 1.
Tracial defects. The identity tau((S - 1)^2) = o(1) is (9.46) written
in the block-weighted trace: sum_beta beta^2 tr_beta((S - 1)^2) is
E ||(S - 1) R_*||_HS^2 up to (1 + epsilon), which Lemma 27 bounds by
o(1). The defect sum_{i, j} tau(|[H_i, H_j]|^2) = o(1) holds because
on the positive block beta the operators H_i commute up to the tau
weight, and operators on different V_beta commute exactly.
Index interchange. The Sylvester relation (9.20) makes the entry
(H_i)_{ab} of the column comparable to the entry (H_a)_{ib}, since
D_i R and D_a R are symmetric in their block indices and the drift is
nondegenerate on each positive block. The commutator estimate (9.21)
controls the residual. Consequently
E sum_beta beta^2 sum_{i, a, b in beta}
|(H_i)_{ab} - (H_a)_{ib}|^2 = o(1). (9.65)
Block-vector action. For each positive block beta and any vector v in
its coordinate range, define
H[v] = sum_{i in beta} v_i H_i. (9.66)
H[.] is the operator whose Gaussian linear combinations produce the
fluctuation matrix of Section 9.9. QED.
9.8 Weighted Gaussian moments.
The block matrices H_i do not commute. The next lemma nevertheless
shows that Gaussian linear combinations D_r = H[N_r] have the scalar
Gaussian joint moments in the tau-weighted limit, using the commutator
and interchange defects of Lemma 28.
Lemma 29. Let the block matrices and tau be those of Lemma 28. For
independent standard Gaussian vectors N_1,...,N_q, independent of the
initial data, put D_r = H[N_r]. Then for every fixed word r_1,...,r_m,
tau(D_{r_1} D_{r_2} ... D_{r_m}) -> E(Z_{r_1} Z_{r_2} ... Z_{r_m}), (9.67)
where Z_1,...,Z_q are independent scalar standard normals. Products of
fixed length can be reordered with error tending to zero in L^2(tau).
For every integer p >= 1 and real u >= 0,
E_{N_r} tr_beta D_r^{2p} <= 4^p (2p - 1)!! tr_beta Id,
tau(e^{u |D_r|}) <= 2 e^{2 u^2} tau(1). (9.68)
If H_j is the tensor Hermite polynomial of degree j, then for each
fixed j,
tau( | sum_{i_1,...,i_j} H_{i_j} ... H_{i_1}
h_j^{(1)}(N_{i_1},...,N_{i_j})
- H[N_r]^j |^2 ) = o(1), (9.69)
where h_j^{(1)} is the scalar Hermite polynomial.
Proof. Schatten-Holder. Define the L^p(tau) norm
||X||_{p,tau} = tau(|X|^p)^{1/p} on the direct sum of the blocks and
average over the classical variables. Schatten Holder followed by
scalar Holder gives
||X_1 ... X_m||_{p,tau}
<= prod_{j=1}^m ||X_j||_{p_j, tau}, sum 1/p_j = 1/p. (9.70)
All bounds use the actual weights beta^2, with no factor depending on a
block dimension. The finite-matrix Schatten inequality follows by the
same singular-value three-lines argument as in Lemma 25, interpolating
endpoint trace and operator norms and applying duality.
Commutator control. For independent N_r, N_s,
||[D_r, D_s]||_{2,tau}^2
= sum_{i, j} tau(|[H_i, H_j]|^2) = o(1). (9.71)
All higher fixed moments are bounded by Holder and (9.70). Interpolation
between L^2(tau) and an arbitrarily high fixed L^p(tau) therefore makes
this commutator tend to zero in every fixed finite power norm. Holder
applied to the factors on either side of a commutator shows that an
adjacent transposition in a word of fixed length costs o(1) in L^2(tau).
A finite sequence of transpositions gives the reordering assertion.
Mixed moments by pairings. Expand a mixed moment
tau(D_{r_1} ... D_{r_m}) by Gaussian pairings, conditional on the
matrices. Pairings must join indices of the same Gaussian vector. For
each pair introduce its own independent standard Gaussian vector,
shared only between the two factors in that pair; this represents the
contracted matrix word exactly as a conditional expectation. Reorder
the word until each pair is adjacent; the error is o(1) even in
L^2(tau). Averaging the independent vectors then gives S^{m/2}, where
S is the conditional square of a single Gaussian matrix.
For each fixed integer b, boundedness of S gives
||S^b - 1||_{2,tau}
<= (sum_{a=0}^{b-1} 4^a) ||S - 1||_{2,tau} = o(1). (9.72)
Each pairing consequently contributes tau(1) + o(1) = 1 + o(1). Odd
moments vanish. The number of admissible pairings is exactly the
scalar independent-normal moment, proving (9.67).
Moment bounds. The first line of (9.68) is Gaussian (2p-1)!! applied
coefficient-wise; the Cauchy-Schwarz factor relating D_r^{2p} to
tr_beta Id is bounded because |H[N_r]|^2 is a quadratic form in N_r
and the various powers of tr_beta Id appear with the fixed coefficient
4^p. The exponential bound follows by summing the moments,
tau(e^{u |D_r|}) = sum_{j >= 0} u^j tau(|D_r|^j)/j!
<= 2 tau(1) sum_{j >= 0} (2 u^2)^j/j! = 2 e^{2 u^2} tau(1).
Wick reduction. Expand the tensor Hermite polynomial using its defining
exponential exp(<v, N> - |v|^2/2). A term consists of free N indices
and a number of signed Kronecker-delta pairs. Represent each delta pair
by its own independent Gaussian vector, and move its two matrix factors
together. Conditional averaging is L^2(tau)-contractive, so the
reordering error remains o(1). The resulting factors S can be replaced
by 1 even between fixed powers of D_1, by Holder and interpolation
applied to the bounded matrix S - 1. The resulting polynomial is
sum_{b=0}^{floor(j/2)} (-1)^b j!/(2^b b! (j - 2 b)!) D_1^{j - 2 b}
= h_j^{(1)}(D_1).
There are only finitely many terms for fixed j, proving (9.69). QED.
9.9 The frozen diffusion law.
Proposition 30 (Frozen diffusion law). Let Y_u be the stationary
diffusion with invariant law eta and driving Brownian motion B_u, and
fix t > 0. Make the matrix construction of Lemma 28 at Y_0 using an
independent random block offset. Set N = B_t / sqrt(t). Conditionally
on the initial data and block offset, N is a standard Gaussian vector.
Define, in the initial spatial coordinates,
sigma = direct_sum_beta beta^2 exp(sqrt(8t) H[N] - 4t Id_beta), (9.73)
on the positive blocks, and sigma = 0 on the omitted kernel. Then
E ||R(Y_t) - sigma^{1/2}||_HS^2 -> 0. (9.74)
The matrices defining sigma obey the bounded-column and input-map bounds
of Lemma 28 and the mixed-moment law of Lemma 29.
Proof. Three steps. Added tensor indices in the hierarchy expansion are
replaced by products of the initial matrices; their Wick contractions
are identified; and the resulting scalar-Hermite series is summed. All
limits involving the degree of that series are taken after the
counterexample-sequence limit, with t fixed.
Step 1. Replacing extra tensor indices. The derivative-action estimate
and hierarchy coupling give
(1/(kI)) sum_{k<=l<2k} sum_i
||T_{l+1, i.} - (H_i)_(1) T_l||_2^2 = o(1). (9.75)
The estimate also holds under a shift of the band by any fixed integer
a >= 0. Removing or adding at most a levels in the error sum costs
O_C(a/k), since the unnormalized squared norm of each one-step error is
at most
2 ||T_{l+1}||_2^2 + 2 C ||T_l||_2^2 = O_C(I), (9.76)
by the bounded column and the endpoint estimates. This also covers the
block and input-clipping modifications already applied.
Iterating (9.75): for each fixed j the extra-index tensor T_{l+j} may
be replaced by
(i_1,...,i_j) -> (H_{i_j})_(1) ... (H_{i_1})_(1) T_l (9.77)
in the normalized band mean-square norm, with error o(1). If e_a
denotes the root mean-square error in the one-step estimate on the band
shifted by a, the root error after j steps is at most
sum_{a=0}^{j-1} C^{a/2} e_{j-1-a}. (9.78)
Multiplication by a column of H_i costs at most sqrt(C) in the summed
coefficient norm. Every e_a = o(1) at fixed a, so the displayed bound
vanishes.
Step 2. Contraction with tensor Hermite. Contraction with the tensor
Hermite polynomial h_j(N) multiplies the squared coefficient norm by at
most j!: it first symmetrizes the contracted indices, an orthogonal
projection, and Gaussian Hermite orthogonality then gives j!.
Combining the Hermite expansion of Section 5 with (9.69), for each
fixed m,
Psi(Y_t) = P_{m,t}(D) Psi(Y_0) + E_{m,t}, D = H[N], (9.79)
where
P_{m,t}(x) = e^{-t} sum_{j=0}^m (2t)^{j/2}/j! h_j^{(1)}(x), (9.80)
limsup ||E_{m,t}||_2 <= (2t)^{(m+1)/2}/sqrt((m+1)!). (9.81)
The band remainder bound follows from the fixed finite-list diagonal at
the start of Section 9. Combined with ||X Psi(Y_0)||_HS = ||X R||_HS
and the matrix field normalization, the Hermite expansion converges in
the tracial polynomial to the operator statement in L^2.
Step 3. Summing the scalar-Hermite series. On every block beta the
field Psi(Y_0) Psi(Y_0)^* = R_*^2 = beta^2 Id_beta, and (9.75) makes H_i
approximately commute with R_*; hence each P_{m,t}(D) acts as a scalar
matrix on block beta at leading order. The generating function of
Hermite polynomials gives
sum_{j=0}^infty (2t)^{j/2}/j! h_j^{(1)}(x)
= e^{x sqrt(2t) - t}. (9.82)
Therefore
P_{m,t}(x) = e^{-t} sum_{j=0}^m (2t)^{j/2}/j! h_j^{(1)}(x)
-> e^{-2t} e^{sqrt(2t) x}. (9.83)
Passing to the operator limit through (9.79),
Psi(Y_t) Psi(Y_t)^*
-> direct_sum_beta beta^2 e^{-4t} e^{sqrt(2t) D} e^{sqrt(2t) D}
= direct_sum_beta beta^2 exp(sqrt(8t) D - 4t Id_beta),
= sigma. (9.84)
Taking the positive square root gives (9.73)-(9.74).
Weak convergence of D. The second moment of D under tau is bounded
(from (9.68) with p = 1), giving tightness. The higher moment bounds
(9.68) give uniform integrability of every polynomial. Every
subsequential limit therefore has the Gaussian moments, and its
exponential moments determine its characteristic function by power
series. Using a larger exponential parameter makes the tails of any
specified test function uniformly small. This completes the
convergence in L^2(HS). QED.
10. THE COVARIANCE OBSERVATION
The frozen diffusion law makes the field R(Y_t) close to a matrix
lognormal of known law. The final contradiction compares two bounds for
the mutual information that a bounded Gaussian observation of R(Y_t)
carries about the diffusion path. One bound comes from the logarithmic
Sobolev inequality of eta. The other comes from the frozen law, via a
scalar Gaussian mixture whose scale integral diverges.
10.1 Setup of the observation.
Fix t > 0 and L >= 1. Define
f_L(q) = q / (1 + q/L) (10.1)
for q >= 0, an increasing bounded function with values in [0, L). Let
C_T(y) = Id + f_L(T R(y)^2) (10.2)
for each scale T > 0. On every positive block beta of R the matrix
C_T(y) acts by multiplication with the scalar
c_beta(T) = 1 + f_L(T beta^2), (10.3)
and on the kernel of R by 1. All eigenvalues of C_T(y) lie in
[1, 1 + L].
For G a standard Gaussian vector in the same coordinates,
independent of the diffusion, the scale-T observation is
Z_T = C_T(Y_t)^{1/2} G. (10.4)
The mixture law of Z_T is ∫ N(0, C_T(y)) dη(y). The information Z_T
carries about Y_t is
MI(Y_t; Z_T) = E H(Law(Y_t | Z_T) | eta). (10.5)
For fixed t, L, T this is finite: the conditional covariance is bounded
above and below, and Gaussian entropy is continuous in its covariance.
10.2 The universal integrated upper bound.
Proposition 31. There is an absolute constant C_0 such that, for
every fixed t > 0 and L >= 1,
limsup ∫_0^infty MI(Y_t; Z_T) dT/T^2 <= C_0. (10.6)
In particular C_0 = 9/4 suffices.
Proof.
Step 1. Likelihood-based LSI bound. Fix y. Write p_T(z | y) for the
centered Gaussian density of covariance C_T(y), p_T(z) = ∫ p_T(z|y) dη(y)
for the marginal, and q_z(y) = p_T(z | y)/p_T(z) for the posterior
density of Y_t given Z_T = z. Applying the LSI of eta to q_z,
2λ H(q_z | eta)
<= I(q_z | eta)
= ∫ |D_y log p_T(z | y)|^2 q_z(y) dη(y). (10.7)
Averaging over z with z ~ p_T(z) and using (10.5),
2λ MI(Y_t; Z_T) <= E |D_y log p_T(Z_T | Y_t)|^2. (10.8)
The expectation is over the joint law (Y_t, Z_T).
Step 2. Score of the Gaussian likelihood. For a positive covariance C
and derivative D_i C, differentiating the centered Gaussian density
in y gives
D_i log p(z | y)
= (1/2)( z^* C^{-1} (D_i C) C^{-1} z - tr(C^{-1} D_i C) ). (10.9)
Conditionally on y, put B_i = C^{-1/2}(D_i C)C^{-1/2} and
g = C^{-1/2} z. Then g is standard Gaussian and
z^* C^{-1} (D_i C) C^{-1} z = g^* B_i g. Expanding the fourth moment
of the standard Gaussian quadratic form,
E[(g^* B_i g - tr B_i)^2] = 2 tr(B_i^2). (10.10)
Summing over i and inserting into (10.8),
2λ MI(Y_t; Z_T)
<= (1/4) E sum_i tr(B_i^2)
= (1/4) E sum_i tr( (C^{-1} D_i C)^2 ). (10.11)
Step 3. Reduction to h_T. Define the scalar function
h_T(v) = sqrt{1 + f_L(T v^2)}, (10.12)
so that C_T(y) = h_T(R(y))^2 by functional calculus (h_T is applied
entrywise on the spectrum of R). Differentiating C = h_T(R)^2 with
respect to y_i and using the chain rule for matrix functions,
D_i C = 2 h_T'(R) h_T(R) D_i R. (10.13)
Since C^{-1} = h_T(R)^{-2},
C^{-1} D_i C = 2 (h_T'(R)/h_T(R)) D_i R. (10.14)
On a positive block beta with R eigenvalue beta,
(h_T'(beta)/h_T(beta))^2
= T beta^2 / ( (1 + T beta^2/L)^2 (1 + (1 + 1/L) T beta^2)^2 ). (10.15)
The right side has the required decay in beta: as beta -> infty, the
denominator grows like (T beta^2/L)^2 (T beta^2)^2 while the numerator
grows like T beta^2, so the ratio decays as beta^{-6}.