diff b1f9bc4..56bc728 spherical_perceptron_bi_orthogonal_v2.txt
diff --git a/spherical_perceptron_bi_orthogonal_v2.txt b/spherical_perceptron_bi_orthogonal_v2.txt
index dc7565d..8073d32 100644
--- a/spherical_perceptron_bi_orthogonal_v2.txt
+++ b/spherical_perceptron_bi_orthogonal_v2.txt
@@ -1,13 +1,13 @@
-Let $A$ be a real $Mtimes N$ matrix, and let $sigma_N$ denote uniform
-probability measure on the sphere $S^N={xinmathbb R^N:bbR^N:|x|^2=N}$.
-For a bounded continuous $f:mathbb RbbRtomathbb R$,bbR$, the normalized
+Let A be a real Mtimes N matrix, and let sigma_N denote uniform
+probability measure on the sphere S^N={xinmathbb R^N:bbR^N:|x|^2=N}.
+For a bounded continuous f:mathbb RbbRtomathbb R,bbR, the normalized
log-partition function of the spherical perceptron is
-$$F_N(A,f)=frac1Nlogint_{S^N}exp{sum_{r=1}^M f((Ax)_r)}dsigma_N(x).$$
+F_N(A,f)=frac1Nlogint_{S^N}exp{sum_{r=1}^M f((Ax)_r)}dsigma_N(x).
-The measurement matrix $A$ carries a disorder distribution on real
-$Mtimes N$ matrices. A temperature parameter enters by writing
-$f=betaphi$. The limit of $F_N(A,f)$ as $M/Ntoalpha>0$ for a
+The measurement matrix A carries a disorder distribution on real
+Mtimes N matrices. A temperature parameter enters by writing
+f=betaphi. The limit of F_N(A,f) as M/Ntoalpha>0 for a
class of disorder laws with bi-orthogonal invariance and prescribed
compact limiting singular spectrum is an explicit variational formula
in which both the activation and the entire limiting spectral
@@ -85,10 +85,10 @@ angular pressure without requiring concavity of the activation.
Gaussian orthogonalisation supplies the underlying Haar frames. The
constructions of Mezzadri are the standard reference; the abstract
states the scope plainly: "random unitary matrices from the classical
-compact groups $U(N)$, $O(N)$ and $USp(2N)$ with probability
+compact groups U(N), O(N) and USp(2N) with probability
distributions given by the respective invariant measures" and the
algorithm is "straightforward to implement using standard linear
-algebra packages." Only the $O(N)$ case is needed.
+algebra packages." Only the O(N) case is needed.
./pdf/arxiv-mezzadri-0609050.pdf
The remaining Gaussian problem involves several Stiefel columns, with
@@ -133,131 +133,131 @@ to a cascade leaf and verifies the marking rules for the fractional
moments and zero restrictions. ./refs/ghirlanda-guerra-1998.pdf
./pdf/panchenko-2013.pdf
-The singular values of $A$ describe the strength and rank of the
-measurements. Write $s_{N,1}ge...ge s_{N,r_N}$ for the singular
-values of $A$, where $r_N=min(M,N)$, with zeros padded in these slots
-when rows or columns are null. Let $delta_a$ denote the Dirac mass at
-$a$, and let $supp,nu$ denote the support of a probability
-measure $nu$ on $[0,∞)$.
-
-Fix $alpha>0$, $M=lflooralpha Nrfloor$, and $fin C_b(mathbb R)$.bbR)$.
-Suppose the law of $A_N$ is unchanged under left and right orthogonal
-multiplication, so $A_Noverset{d}{=}UA_NV$ for any independent
-uniform orthogonal $Uin O(M)$, $Vin O(N)$, and suppose the empirical
-singular-value measure $r_N^{-1}sum_{rle r_N}delta_{s_{N,r}}$
+The singular values of A describe the strength and rank of the
+measurements. Write s_{N,1}ge...ge s_{N,r_N} for the singular
+values of A, where r_N=min(M,N), with zeros padded in these slots
+when rows or columns are null. Let delta_a denote the Dirac mass at
+a, and let supp,nu denote the support of a probability
+measure nu on [0,∞).
+
+Fix alpha>0, M=lflooralpha Nrfloor, and fin C_b(mathbb R).bbR).
+Suppose the law of A_N is unchanged under left and right orthogonal
+multiplication, so A_Noverset{d}{=}UA_NV for any independent
+uniform orthogonal Uin O(M), Vin O(N), and suppose the empirical
+singular-value measure r_N^{-1}sum_{rle r_N}delta_{s_{N,r}}
converges weakly in probability to a deterministic compactly supported
-probability measure $nu$. Suppose in addition that
-$max_{rle r_N}dist(s_{N,r},supp,nu)to 0$ in
-probability. Then $F_N(A_N,f)$ converges in probability and in mean to
-a deterministic value $F_{alpha,nu}(f)$.
+probability measure nu. Suppose in addition that
+max_{rle r_N}dist(s_{N,r},supp,nu)to 0 in
+probability. Then F_N(A_N,f) converges in probability and in mean to
+a deterministic value F_{alpha,nu}(f).
-The formula for $F_{alpha,nu}(f)$ splits into a radial part driven by
+The formula for F_{alpha,nu}(f) splits into a radial part driven by
how the spherical input decomposes among the singular-value blocks, and
an angular part kept inside an auxiliary Gaussian model. The radial part
is a Dirichlet optimisation; the angular part is a matrix-path
variational problem evaluated by the Gaussian theorem.
Approximate the singular slots in operator norm by finitely many
-positive values $s_1,...,s_h$ with multiplicities $n_j$ satisfying
-$n_j/Mto p_j>0$. Then $sum_jp_j=min(1,1/alpha)$. Put
-$v_j=alpha p_j$, and when $alpha<1$ add an input null block with
-$v_0=1-alpha$. Let $mathcal I=calI={1,...,h}$ when $alphage 1$ and
-$mathcal I=calI={0,1,...,h}$ when $alpha<1$. For each block
-$iinmathcal I$calI$ let $rho_i$ denote the fraction of the input squared
-norm assigned to block $i$, with $rho_i>0$ and $sum_irho_i=1$. The
+positive values s_1,...,s_h with multiplicities n_j satisfying
+n_j/Mto p_j>0. Then sum_jp_j=min(1,1/alpha). Put
+v_j=alpha p_j, and when alpha<1 add an input null block with
+v_0=1-alpha. Let mathcal I=calI={1,...,h} when alphage 1 and
+mathcal I=calI={0,1,...,h} when alpha<1. For each block
+iinmathcal IcalI let rho_i denote the fraction of the input squared
+norm assigned to block i, with rho_i>0 and sum_irho_i=1. The
value for this finite approximation is
-$$sup_{rho}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)},$$
+sup_{rho}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)},
-where $P_p^f$ is the angular pressure characterised by the Gaussian theorem and
-$p=(p_j)_j$ fixes every block proportion. The limit of these
-approximations as the spectral mesh tends to zero is $F_{alpha,nu}(f)$.
+where P_p^f is the angular pressure characterised by the Gaussian theorem and
+p=(p_j)_j fixes every block proportion. The limit of these
+approximations as the spectral mesh tends to zero is F_{alpha,nu}(f).
-The term $frac12sum v_ilog(rho_i/v_i)$ is the logarithmic Dirichlet
+The term frac12sum v_ilog(rho_i/v_i) is the logarithmic Dirichlet
density, at rate per input coordinate, for distributing the squared
length of a spherically uniform input across blocks of dimensions
-$v_iN$. The factor $alpha$ converts the per-$mathbb R^M$bbR^M$ angular
-pressure into a per-$mathbb R^N$bbR^N$ pressure.
-
-Let $E_1,...,E_hsubsetmathbb R^M$bbR^M$ be mutually orthogonal Haar
-subspaces with $dim E_j=n_j$ and $n_j/Mto p_j>0$, so $p=sum_jp_jle 1$.
-For $a_jge 0$, let $sigma_{E_j,Ma_j}$ denote uniform probability
-measure on the sphere ${yin E_j:|y|^2=Ma_j}$ of squared radius
-$Ma_j$, with the point mass at $y=0$ when $a_j=0$. Define the random
+v_iN. The factor alpha converts the per-mathbb R^MbbR^M angular
+pressure into a per-mathbb R^NbbR^N pressure.
+
+Let E_1,...,E_hsubsetmathbb R^MbbR^M be mutually orthogonal Haar
+subspaces with dim E_j=n_j and n_j/Mto p_j>0, so p=sum_jp_jle 1.
+For a_jge 0, let sigma_{E_j,Ma_j} denote uniform probability
+measure on the sphere {yin E_j:|y|^2=Ma_j} of squared radius
+Ma_j, with the point mass at y=0 when a_j=0. Define the random
angular pressure
-$$P_M^f(a)=frac1Mlogintexp{sum_{r=1}^M f(sum_jy_{j,r})}prod_j dsigma_{E_j,Ma_j}(y_j).$$
+P_M^f(a)=frac1Mlogintexp{sum_{r=1}^M f(sum_jy_{j,r})}prod_j dsigma_{E_j,Ma_j}(y_j).
-The subspaces are held fixed inside the integral, and the vectors $y_j$
+The subspaces are held fixed inside the integral, and the vectors y_j
are integrated independently. The angular theorem asserts that
-$P_M^f(a)$ converges in probability, uniformly on compact sets of
-$ain[0,∞)^h$, to a deterministic continuous function $P_p^f(a)$.
-For $p<1$ and positive $a$, the limit equals
+P_M^f(a) converges in probability, uniformly on compact sets of
+ain[0,∞)^h, to a deterministic continuous function P_p^f(a).
+For p<1 and positive a, the limit equals
-$$P_p^f(a)=J_p-H_p(a)+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta),$$
+P_p^f(a)=J_p-H_p(a)+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta),
-for every sufficiently large fixed $kappa$, with
+for every sufficiently large fixed kappa, with
-$$s_a=sum_ja_j,qquad H_p(a)=frac12sum_jp_jlog(2pi e a_j)/(p_j),qquad J_p=frac12int_0^plog(1-u),du.$$
+s_a=sum_ja_j, H_p(a)=frac12sum_jp_jlog(2pi e a_j)/(p_j), J_p=frac12int_0^plog(1-u),du.
-The quantity $H_p$ is the entropy per output coordinate of a product of
-Euclidean shells of radii $a_j$, and $J_p$ is the logarithmic Jacobian
-of the Gaussian span map. Neither depends on $f$. For the trivial
-activation $f=0$, the two combine with the Gaussian part to give
-$P_p^0(a)=0$, matching $P_M^0(a)=0$. Values at zero coordinates of $a$
-are obtained by continuity. When $sum_jp_j=1$,
-$P_p^f(a)=lim_{eta↓ 0}P_{(1-eta)p}^f(a)$ with $a$ fixed.
+The quantity H_p is the entropy per output coordinate of a product of
+Euclidean shells of radii a_j, and J_p is the logarithmic Jacobian
+of the Gaussian span map. Neither depends on f. For the trivial
+activation f=0, the two combine with the Gaussian part to give
+P_p^0(a)=0, matching P_M^0(a)=0. Values at zero coordinates of a
+are obtained by continuity. When sum_jp_j=1,
+P_p^f(a)=lim_{eta↓ 0}P_{(1-eta)p}^f(a) with a fixed.
-Let $d_j=h-j+1$. For each $j$, let $R_j$ be an upper triangular
-$d_jtimes d_j$ matrix with positive diagonal and every column norm at
-most $kappa$. For a row vector $xi_jinmathbb R^bbR^{d_j}$, label the
-entries of $xi_j^TR_j$ by $x,j$ and $z_i,j$ for $jle i<h$. Write
+Let d_j=h-j+1. For each j, let R_j be an upper triangular
+d_jtimes d_j matrix with positive diagonal and every column norm at
+most kappa. For a row vector xi_jinmathbb R^bbR^{d_j}, label the
+entries of xi_j^TR_j by x,j and z_i,j for jle i<h. Write
-$$y=sum_j(xi_j^TR_j)_{x,j},qquad u_i=sum_{jle i}(xi_j^TR_j)_{z_i,j}.$$
+y=sum_j(xi_j^TR_j)_{x,j}, u_i=sum_{jle i}(xi_j^TR_j)_{z_i,j}.
-The labels $y,u_i$ refer to single coordinates of the row fields in the
+The labels y,u_i refer to single coordinates of the row fields in the
Gaussian model. Set
-$$g_1=y^2, c_1=s_a;qquad g_2=-min(y^2,B), c_2=-s_a;qquad g_{i+2}=(y-u_i)^2, c_{i+2}=sum_{j>i}a_j.$$
+g_1=y^2, c_1=s_a; g_2=-min(y^2,B), c_2=-s_a; g_{i+2}=(y-u_i)^2, c_{i+2}=sum_{j>i}a_j.
Define
-$$mathcal B_calB_{p,a}(kappa,lambda,B,theta)=sup_{R_1,...,R_h}[sum_jp_j(log(kappa√((2pi e)/(p_j)))+logfrac{R_{j,11}}kappa)+thetasum_{ell=2}^{d_j}p_jlogfrac{R_{j,ellell}}kappa+thetamin_{0le t_elllelambda/theta}{tcdot c+mathcal G_calG_{theta,p}((f(y))/(theta)-tcdot g)}],$$
+mathcal B_calB_{p,a}(kappa,lambda,B,theta)=sup_{R_1,...,R_h}[sum_jp_j(log(kappa√((2pi e)/(p_j)))+logfrac{R_{j,11}}kappa)+thetasum_{ell=2}^{d_j}p_jlogfrac{R_{j,ellell}}kappa+thetamin_{0le t_elllelambda/theta}{tcdot c+mathcal G_calG_{theta,p}((f(y))/(theta)-tcdot g)}],
-where $mathcal G_calG_{theta,p}$ is the Gaussian functional obtained from
+where mathcal G_calG_{theta,p} is the Gaussian functional obtained from
the powered Stiefel integral. The first column of
-$R_j$ has weight $p_j$ and fires the outer entropy. The remaining
-columns have weight $p_jtheta$ and the entropy of those columns
-vanishes as $theta↓ 0$. The min over $t$ is a Lagrangian dual for the
-row-average constraints $M^{-1}sum_r g_ell(y_r,...)le c_ell$ in
-the limit $lambdato∞$. The truncation $B$ is removed last.
+R_j has weight p_j and fires the outer entropy. The remaining
+columns have weight p_jtheta and the entropy of those columns
+vanishes as theta↓ 0. The min over t is a Lagrangian dual for the
+row-average constraints M^{-1}sum_r g_ell(y_r,...)le c_ell in
+the limit lambdato∞. The truncation B is removed last.
The auxiliary Gaussian model distinguishes the vector whose weight is
sought from the vectors that witness its projection constraints. Fix
-arbitrary column counts $d_jge 1$ and a positive exponent
-$0<theta<1$. Let $n_j/Mto p_j>0$ and let
+arbitrary column counts d_jge 1 and a positive exponent
+0<theta<1. Let n_j/Mto p_j>0 and let
-$$X_jinSt(n_j,d_j):={Xinmathbb R^bbR^{n_jtimes d_j}:X^TX=n_jI_{d_j}}$$
+X_jinSt(n_j,d_j):={Xinmathbb R^bbR^{n_jtimes d_j}:X^TX=n_jI_{d_j}}
be uniform on the Stiefel manifold. The first columns, jointly across
species, are outer variables; the remaining columns have the
conditional Stiefel law and are inner variables. For independent
-standard Gaussian vectors $g_{r,j}inmathbb R^bbR^{n_j}$, define the row
+standard Gaussian vectors g_{r,j}inmathbb R^bbR^{n_j}, define the row
fields
-$$xi_{r,j}(X)=frac{X_j^Tg_{r,j}}{√(n_j)},qquad 1le rle M.$$
+xi_{r,j}(X)=frac{X_j^Tg_{r,j}}{√(n_j)}, 1le rle M.
-For a Hamiltonian $H$ define
+For a Hamiltonian H define
-$$mathcal Z_calZ_{M,theta}(H)=[int(int e^H,dmu_{in})^theta dmu_{out}]^{1/theta}.$$
+mathcal Z_calZ_{M,theta}(H)=[int(int e^H,dmu_{in})^theta dmu_{out}]^{1/theta}.
The natural tilted path law samples the outer variables with density
-proportional to the inner partition function to power $theta$, and then
+proportional to the inner partition function to power theta, and then
samples the tilted conditional inner law.
The first column is drawn once before the inner columns are integrated.
In the cascade representation it is shared by visits whose common level
-is at least $theta$; the other columns remain visit variables. The
+is at least theta; the other columns remain visit variables. The
matrix paths record the covariances of Gaussian row
fields seen by such visits. In the Gaussian theorem those matrix paths
determine a variational functional whose value matches the limit of the
@@ -265,69 +265,69 @@ normalized log integral.
Choose a common finite grid
-$$0=zeta_0<zeta_1<...<zeta_k=1,qquad w_i=zeta_{i+1}-zeta_i,quad 0le i<k,$$
+0=zeta_0<zeta_1<...<zeta_k=1, w_i=zeta_{i+1}-zeta_i, 0le i<k,
-containing $theta$. A trial in block $j$ consists of symmetric matrices
+containing theta. A trial in block j consists of symmetric matrices
-$$0preceq q_{j,0}preceq...preceq q_{j,k-1}preceq I_{d_j},qquad q_{j,-1}=0,quad q_{j,k}=I_{d_j},$$
+0preceq q_{j,0}preceq...preceq q_{j,k-1}preceq I_{d_j}, q_{j,-1}=0, q_{j,k}=I_{d_j},
-with the pinning condition $q_{j,i}e_1=e_1$ whenever $zeta_igetheta$.
-Write $q_j(u)=q_{j,i}$ on $[zeta_i,zeta_{i+1})$. Let
-$F_{j,i}=mathbb R^bbR^{d_j}$ before the sharing level and
-$F_{j,i}=e_1^perp$ afterwards, and put $F_{j,k}={0}$. The
+with the pinning condition q_{j,i}e_1=e_1 whenever zeta_igetheta.
+Write q_j(u)=q_{j,i} on [zeta_i,zeta_{i+1}). Let
+F_{j,i}=mathbb R^bbR^{d_j} before the sharing level and
+F_{j,i}=e_1^perp afterwards, and put F_{j,k}={0}. The
susceptibility and precision are
-$$mathsf K_sfK_{j,i}=sum_{r=i+1}^kzeta_r(q_{j,r}-q_{j,r-1}^{phantom{}}),qquad mathsf P_sfP_{j,i}=(mathsf K_sfK_{j,i}|_{F_{j,i}})^{-1}.$$
+mathsf K_sfK_{j,i}=sum_{r=i+1}^kzeta_r(q_{j,r}-q_{j,r-1}^{phantom{}}), mathsf P_sfP_{j,i}=(mathsf K_sfK_{j,i}|_{F_{j,i}})^{-1}.
An eligible trial has strictly positive susceptibility on every
nonempty active space. The precision extends by zero on the orthogonal
-complement. Let $q_j^0$ be zero on active coordinates and the identity
-on shared coordinates, with $q_{j,k}^0=I$, and define
-$mathsf K_sfK_{j,i}^0$ analogously. The block entropy is
+complement. Let q_j^0 be zero on active coordinates and the identity
+on shared coordinates, with q_{j,k}^0=I, and define
+mathsf K_sfK_{j,i}^0 analogously. The block entropy is
-$$2S_theta(q_j)=tr(mathsf P_sfP_{j,0}q_{j,0})+sum_{i=1}^kfrac1{zeta_i}log(frac{detmathsf K_sfK_{j,i-1}}{detmathsf K_sfK_{j,i}}frac{detmathsf K_sfK_{j,i}^0}{detmathsf K_sfK_{j,i-1}^0}),$$
+2S_theta(q_j)=tr(mathsf P_sfP_{j,0}q_{j,0})+sum_{i=1}^kfrac1{zeta_i}log(frac{detmathsf K_sfK_{j,i-1}}{detmathsf K_sfK_{j,i}}frac{detmathsf K_sfK_{j,i}^0}{detmathsf K_sfK_{j,i-1}^0}),
where each determinant is taken on the active space at its own index.
-The normalisation compensates for the dimension change at $theta$.
+The normalisation compensates for the dimension change at theta.
-For a continuous terminal $D:prod_jmathbb R^bbR^{d_j}tomathbb R$bbR$
-satisfying $-C(1+|z|^2)le D(z)le C$ for some $C<∞$, let
-$Z_0,...,Z_k$ be independent centred Gaussian vectors with
+For a continuous terminal D:prod_jmathbb R^bbR^{d_j}tomathbb RbbR
+satisfying -C(1+|z|^2)le D(z)le C for some C<∞, let
+Z_0,...,Z_k be independent centred Gaussian vectors with
block-diagonal covariances
-$diag_j(q_{j,i}-q_{j,i-1})$. Set $U_k=D$ and define backwards,
-for $i=k,k-1,...,1$,
+diag_j(q_{j,i}-q_{j,i-1}). Set U_k=D and define backwards,
+for i=k,k-1,...,1,
-$$U_{i-1}(z)=frac1{zeta_i}logmathbb EbbEexp{zeta_iU_i(z+Z_i)}.$$
+U_{i-1}(z)=frac1{zeta_i}logmathbb EbbEexp{zeta_iU_i(z+Z_i)}.
-The single-pattern value is $V_D(q)=mathbb EU_0(Z_0)$;bbEU_0(Z_0)$; the outer
+The single-pattern value is V_D(q)=mathbb EU_0(Z_0);bbEU_0(Z_0); the outer
expectation averages the root Gaussian data. Define
-$$mathcal G_calG_{theta,p}(D)=inf_q{sum_{j=1}^hp_jS_theta(q_j)+V_D(q)},$$
+mathcal G_calG_{theta,p}(D)=inf_q{sum_{j=1}^hp_jS_theta(q_j)+V_D(q)},
over all eligible finite grids and trials. Refinements use a common
-grid for all blocks. The diagonal increment $I-q_{j,k-1}$ is integrated
+grid for all blocks. The diagonal increment I-q_{j,k-1} is integrated
independently on every visit in the cascade interpretation, including
repeated visits to the same leaf.
The Gaussian theorem states that, with the Stiefel model, fixed
-$0<theta<1$, $n_j/Mto p_j>0$, and $D$ bounded continuous or the sum
+0<theta<1, n_j/Mto p_j>0, and D bounded continuous or the sum
of a bounded continuous function and a nonpositive quadratic form, the
normalized log integral satisfies
-$$frac1Mlogmathcal Z_calZ_{M,theta}(sum_{r=1}^MD(xi_r))→mathcal G_calG_{theta,p}(D)$$
+frac1Mlogmathcal Z_calZ_{M,theta}(sum_{r=1}^MD(xi_r))→mathcal G_calG_{theta,p}(D)
-in probability and in expectation. More generally, let each $g_ell$
+in probability and in expectation. More generally, let each g_ell
be bounded continuous or a nonnegative quadratic form, let
-$barbar g=M^g=M^{-1}sum_rg(xi_r)$, and fix $d>0$ and $cinmathbb R^s$.bbR^s$.
+barbar g=M^g=M^{-1}sum_rg(xi_r), and fix d>0 and cinmathbb R^s.bbR^s.
The pressure for the soft-penalised Hamiltonian
-$$H=sum_rD(xi_r)-Mdsum_{ell=1}^s(barbar g_g_ell-c_ell)_+$$
+H=sum_rD(xi_r)-Mdsum_{ell=1}^s(barbar g_g_ell-c_ell)_+
converges in probability to
-$$min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.$$
+min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.
-No restriction on $sum_jp_j$ is needed.
+No restriction on sum_jp_j is needed.
The terminal class "bounded continuous or bounded continuous plus a
nonpositive quadratic" covers every terminal that occurs in the
@@ -335,47 +335,47 @@ construction. The base bounded case is trivial for the estimates. The
negative quadratic form is the shape produced when a shrinkage penalty
enforces bounded squared norm. A positive quadratic would increase
exponentially with the field and break the variational formula, so the
-sign restriction excludes exactly that case. In $mathcal B_calB_{p,a}$ the
-row tests $g_ell$ are nonnegative quadratics, and the contributions
-$-tcdot g$ fall in the negative-quadratic class.
+sign restriction excludes exactly that case. In mathcal B_calB_{p,a} the
+row tests g_ell are nonnegative quadratics, and the contributions
+-tcdot g fall in the negative-quadratic class.
Reparametrise the powered integral as the partition function of a
branching marked cascade. Choose a Poisson point process on the
-$k$-level tree with leaf-measure normalised to one per generation, and
+k-level tree with leaf-measure normalised to one per generation, and
give each leaf an independent copy of the inner Stiefel law. The
outer first-column mark at a branching node is drawn at exponent
-$theta$; the visit variables attached to the same root share that
+theta; the visit variables attached to the same root share that
mark. Reads on inner visits are independent at every occurrence,
including repeated reads from the same leaf.
Under this construction, the expected cascade logarithm conditioned on
-disorder equals $(1/theta)logint(int e^Hdmu_{rm in})^theta dmu_{rm out}$.
-When $H$ has no further label dependence, adding redundant cascade
+disorder equals (1/theta)logint(int e^Hdmu_{rm in})^theta dmu_{rm out}.
+When H has no further label dependence, adding redundant cascade
levels leaves this value unchanged. Merged masses at a branching
-exponent $z$ follow normalised stable jumps at exponent $z$; at the
-fixed exponent $theta$ this gives cluster masses that are stable
+exponent z follow normalised stable jumps at exponent z; at the
+fixed exponent theta this gives cluster masses that are stable
jumps independent of finer subdivision. Conditional inner Stiefel
limits at each mark are as in the single-root case: spatial frames
converge to uniform Stiefel points, and read covariances are the
-paths $q$.
+paths q.
-Because $mathcal Z_calZ_{M,theta}(H)$ is a normalised log of a partition
+Because mathcal Z_calZ_{M,theta}(H) is a normalised log of a partition
function over independent patterns and independent disorder, standard
subadditivity and Gaussian concentration bound the variance of
-$M^{-1}logmathcal Z_calZ_{M,theta}(H)$ by $O(M^{-1})$. Consequently the
+M^{-1}logmathcal Z_calZ_{M,theta}(H) by O(M^{-1}). Consequently the
normalised log concentrates at its mean, and convergence in probability
and convergence in expectation are equivalent.
Overlaps and matrix synchronization. Write replicas as superscripts
-$a,b$. The scalar and matrix overlaps of a configuration $(X,A)$ at
+a,b. The scalar and matrix overlaps of a configuration (X,A) at
the disorder level are
-$$mathcal R_j^calR_j^{ab}=((X_j^a)^TX_j^b)/(n_j).$$
+mathcal R_j^calR_j^{ab}=((X_j^a)^TX_j^b)/(n_j).
The array indexed jointly by replica and column is Gram and the self
-blocks are fixed at $I$. The perturbing scalarizations are
-$v^T(mathcal R_j^calR_j^{ab})^{circell}v$ for $ell=1,2$ and a countable
-dense set of vectors $v$, together with the numerical label overlap.
+blocks are fixed at I. The perturbing scalarizations are
+v^T(mathcal R_j^calR_j^{ab})^{circell}v for ell=1,2 and a countable
+dense set of vectors v, together with the numerical label overlap.
Each such scalarization is a Gram kernel: take tensor products of the
column vectors for the entrywise powers. Products, nonnegative
integral powers, and positive sums remain Gram kernels. After
@@ -384,17 +384,17 @@ diagonals are constant.
For a Hamiltonian with these joint scalarizations added and small
Gaussian perturbations imposed on each, the Ghirlanda-Guerra identities
-hold in the limit. Precisely, for every bounded test $F$ of the
-complete arrays on the first $r$ replicas and every continuous test
-$psi$ of the generated scalarization-and-label vector on one edge,
+hold in the limit. Precisely, for every bounded test F of the
+complete arrays on the first r replicas and every continuous test
+psi of the generated scalarization-and-label vector on one edge,
-$$mathbb E<bbE< Fpsi(T^{1,r+1})>=frac1rmathbb E<bbE< F>,mathbb E<bbE<psi(T^{12})>+frac1rsum_{a=2}^rmathbb E<bbE< Fpsi(T^{1a})>,$$
+mathbb E<bbE< Fpsi(T^{1,r+1})>=frac1rmathbb E<bbE< F>,mathbb E<bbE<psi(T^{12})>+frac1rsum_{a=2}^rmathbb E<bbE< Fpsi(T^{1a})>,
-where brackets denote Gibbs expectation and $mathbb E$bbE$ averages all
+where brackets denote Gibbs expectation and mathbb EbbE averages all
randomness. Monomial GG identities give all continuous tests of this
generated vector by polynomial approximation on its compact range.
At this stage the identities do not yet identify a nonsymmetric
-matrix from its scalarizations; the test $F$ may nevertheless depend
+matrix from its scalarizations; the test F may nevertheless depend
on the complete finite matrix arrays.
Symmetry of the matrix overlap in the limit follows from the GG
@@ -402,11 +402,11 @@ identities and Panchenko's ultrametricity theorem for scalarized
coordinates. A countable separating family of first-power quadratic
forms is comonotone on an edge; summing over a countable family with
positive weights gives a single order coordinate. If the matrix
-overlap $mathcal R_j^calR_j^{12}$ were not symmetric in the limit, dense
-first-power forms $alpha$ and antisymmetric perturbations $beta$ of
-$mathcal R_j^calR_j^{12}$ would generate crossed values on edges $12$ and
-$13$ that the GG and ultrametricity structure forbid. Ultrametricity
-of each of two crossed coordinates forces edge $23$ to have the lower
+overlap mathcal R_j^calR_j^{12} were not symmetric in the limit, dense
+first-power forms alpha and antisymmetric perturbations beta of
+mathcal R_j^calR_j^{12} would generate crossed values on edges 12 and
+13 that the GG and ultrametricity structure forbid. Ultrametricity
+of each of two crossed coordinates forces edge 23 to have the lower
value in each coordinate, contradicting ultrametricity of the sum.
The required form of the ultrametric summary is the version proved by
Panchenko for vector spins and for the Potts spin glass, together
@@ -420,152 +420,152 @@ Sznitman.
Matrix quantiles. Dense first-power quadratic forms determine each
matrix once symmetry is known. Dense quadratic forms therefore give
-deterministic matrix quantiles $q_j(u)$, nondecreasing in Loewner
-order. Gram positivity and the fixed self overlap $I$ imply
-$0preceq q_jpreceq I$. The label coordinate retains its prescribed
-atom masses, so its sharing threshold sits at quantile $theta$. When
-the first column is shared, $mathcal R_j^calR_j^{ab}e_1=e_1$; hence the
+deterministic matrix quantiles q_j(u), nondecreasing in Loewner
+order. Gram positivity and the fixed self overlap I imply
+0preceq q_jpreceq I. The label coordinate retains its prescribed
+atom masses, so its sharing threshold sits at quantile theta. When
+the first column is shared, mathcal R_j^calR_j^{ab}e_1=e_1; hence the
matrix quantiles satisfy the pinning condition at that threshold.
-Variation of the single-pattern value. For smooth $D$, the functional
-$V_D$ is Lipschitz in the sum of the $L^1$ path distances. Along a
+Variation of the single-pattern value. For smooth D, the functional
+V_D is Lipschitz in the sum of the L^1 path distances. Along a
step variation,
-$$left.(d)/(dt)V_D(q+tdot q)|_{t=0}=-frac12sum_{j,i}w_i,tr(dot q_{j,i}mathbb E[M_bbE[M_{j,i}M_{j,i}^T]),$$
+left.(d)/(dt)V_D(q+tdot q)|_{t=0}=-frac12sum_{j,i}w_i,tr(dot q_{j,i}mathbb E[M_bbE[M_{j,i}M_{j,i}^T]),
-where $M_{j,i}$ is the conditional mean of the terminal $j$-gradient
+where M_{j,i} is the conditional mean of the terminal j-gradient
under the successive tilted kernels. Gaussian covariance
differentiation inserts half the Hessian and the step coefficient
times the gradient square; propagating these through the cascade makes
the Hessian terms telescope, and consecutive coefficients differ by
-$w_i$. The conditional gradients form a martingale, so the second
-moment matrices $G_{j,i}=mathbb E[M_bbE[M_{j,i}M_{j,i}^T]$ are nondecreasing
-and uniformly bounded. Put $T_{j,i}=sum_{r=i}^{k-1}w_r(widetilde
-q_{j,r}-q_{j,r})$ and $T_{j,k}=0$. Summation by parts gives
+w_i. The conditional gradients form a martingale, so the second
+moment matrices G_{j,i}=mathbb E[M_bbE[M_{j,i}M_{j,i}^T] are nondecreasing
+and uniformly bounded. Put T_{j,i}=sum_{r=i}^{k-1}w_r(widetilde
+q_{j,r}-q_{j,r}) and T_{j,k}=0. Summation by parts gives
-$$sum_iw_itr((widetilde q_{j,i}-q_{j,i})G_{j,i})=tr(T_{j,0}G_{j,0})+sum_{i=1}^{k-1}tr(T_{j,i}(G_{j,i}-G_{j,i-1})),$$
+sum_iw_itr((widetilde q_{j,i}-q_{j,i})G_{j,i})=tr(T_{j,0}G_{j,0})+sum_{i=1}^{k-1}tr(T_{j,i}(G_{j,i}-G_{j,i-1})),
and every term is nonnegative when the tail differences are positive
semidefinite. Therefore
-$$int_t^1widetilde q_j(u)dusucceqint_t^1q_j(u)duquadhbox{for every }j,t ⇒ V_D(widetilde q)le V_D(q).$$
+int_t^1widetilde q_j(u)dusucceqint_t^1q_j(u)duquadhbox{for every }j,t ⇒ V_D(widetilde q)le V_D(q).
The identity and the last implication are used in the upper bound.
The canonical lemma. Consider one block, suppressing its index. Fix a
-grid and covariance levels $0preceq b_0preceq... preceq b_{k-1}$
+grid and covariance levels 0preceq b_0preceq... preceq b_{k-1}
of a Gaussian linear field, distinct from the spin-covariance levels
-$q$. Let $ell_n(b)$ denote the expected cascade log integral per
+q. Let ell_n(b) denote the expected cascade log integral per
spatial coordinate for the Stiefel model and this linear field, after
-subtracting $(1/2)tr,b_{k-1}$. Then
+subtracting (1/2)tr,b_{k-1}. Then
-$$limsup_{nto∞}ell_n(b)le S_theta(q)-frac12sum_iw_itr(q_ib_i)$$
+limsup_{nto∞}ell_n(b)le S_theta(q)-frac12sum_iw_itr(q_ib_i)
-for every eligible $q$, locally uniformly in the field levels.
+for every eligible q, locally uniformly in the field levels.
The proof identifies the right side as the value of a dual problem
-whose minimiser is constructed by Gaussian tilting. Let $C=C(b)$ range
-over symmetric $dtimes d$ multipliers. Define
-
-$$Psi(C,b)=frac12tr(Cb_0^{phantom{.}})+hbox{(expected tilted energy functionals)}$$
-
-for the tilted spin density $N(0,Q)$ and the tilted kernels attached
-to $C$. The exact expression is not needed; only its structure
-matters. This $Psi(C,b)$ is lower semicontinuous in $C$ with value
-$+∞$ outside its finite domain, and has a coercivity bound.
-Indeed for any covariance $Q$ with $tfrac12Ipreceq Qpreceqtfrac32 I$,
-use $N(0,Q)$ as trial in the spin integration leaving the field
-kernels unchanged. Write $gamma_r=N(0,I_r)$ and $Q_1,Q_{Fmid1}$ for
+whose minimiser is constructed by Gaussian tilting. Let C=C(b) range
+over symmetric dtimes d multipliers. Define
+
+Psi(C,b)=frac12tr(Cb_0^{phantom{.}})+hbox{(expected tilted energy functionals)}
+
+for the tilted spin density N(0,Q) and the tilted kernels attached
+to C. The exact expression is not needed; only its structure
+matters. This Psi(C,b) is lower semicontinuous in C with value
++∞ outside its finite domain, and has a coercivity bound.
+Indeed for any covariance Q with tfrac12Ipreceq Qpreceqtfrac32 I,
+use N(0,Q) as trial in the spin integration leaving the field
+kernels unchanged. Write gamma_r=N(0,I_r) and Q_1,Q_{Fmid1} for
the marginal and conditional laws of this trial. The chain rule for
relative entropy gives
-$$theta^{-1}Ent(Q_1midgamma_1)+mathbb E_bbE_{Q_1}Ent(Q_{Fmid1}midgamma_{d-1})letheta^{-1}Ent(N(0,Q)midgamma_d),$$
+theta^{-1}Ent(Q_1midgamma_1)+mathbb E_bbE_{Q_1}Ent(Q_{Fmid1}midgamma_{d-1})letheta^{-1}Ent(N(0,Q)midgamma_d),
and this permits correlated trial coordinates. Hence
-$$Psi(C,b)-tfrac12tr,Cgetfrac12tr(C(Q-I))-frac1{2theta}(tr,Q-logdet Q-d).$$
+Psi(C,b)-tfrac12tr,Cgetfrac12tr(C(Q-I))-frac1{2theta}(tr,Q-logdet Q-d).
-Choosing $Q=I+tfrac12sign(C)$ yields
+Choosing Q=I+tfrac12sign(C) yields
-$$Psi(C,b)-tfrac12tr,Cgetfrac14|C|_*-c_d/theta,$$
+Psi(C,b)-tfrac12tr,Cgetfrac14|C|_*-c_d/theta,
-with $|cdot|_*$ the trace norm. When $|b_{k-1}|le B_0$, the
-comparison constant $C=-(B_0+1)I$ gives $Psi(C,b)le 0$. Nested
-exponents dominate $1$ and Jensen's bound controls the ordinary
+with |cdot|_* the trace norm. When |b_{k-1}|le B_0, the
+comparison constant C=-(B_0+1)I gives Psi(C,b)le 0. Nested
+exponents dominate 1 and Jensen's bound controls the ordinary
Gaussian exponential integral. Thus minimisers of
-$Psi(C,b)-tr,C/2$ lie in a compact set depending only on
-$d,theta,B_0$. Lower semicontinuity gives existence; finiteness
-forces strict precision inequalities, and differentiating in $C$
-gives the tilted self-covariance identity $mathbb Ess^T=I$.bbEss^T=I$.
-
-Under the tilted kernels, let $mathcal H_i$calH_i$ be the sigma-field
-generated by the field increments up to $zeta_i$ plus the shared spin
-coordinates when $zeta_igetheta$, and let $M_i=mathbb E[sbbE[smid
-mathcal H_i]$.calH_i]$. Let $mathsf K_i^*$sfK_i^*$ denote the Hessian of $U_i$ in its
-field argument, supported on the active space $F_i$. Differentiating a
+Psi(C,b)-tr,C/2 lie in a compact set depending only on
+d,theta,B_0. Lower semicontinuity gives existence; finiteness
+forces strict precision inequalities, and differentiating in C
+gives the tilted self-covariance identity mathbb Ess^T=I.bbEss^T=I.
+
+Under the tilted kernels, let mathcal H_icalH_i be the sigma-field
+generated by the field increments up to zeta_i plus the shared spin
+coordinates when zeta_igetheta, and let M_i=mathbb E[sbbE[smid
+mathcal H_i].calH_i]. Let mathsf K_i^*sfK_i^* denote the Hessian of U_i in its
+field argument, supported on the active space F_i. Differentiating a
single log-exponential integration gives the recursion
-$$mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_iCov(M_imidmathcal H_calH_{i-1}),$$
+mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_iCov(M_imidmathcal H_calH_{i-1}),
with deterministic conditional covariances. Averaging and the
martingale property yield
-$$q_i^*=mathbb E[M_iM_i^T],bbE[M_iM_i^T],qquad mathsf K_k^*=0,sfK_k^*=0,qquad mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_i(q_i^*-q_{i-1}^*),qquad q_0^*=mathsf K_0^*b_0sfK_0^*b_0mathsf K_0^*.$$sfK_0^*.$$
+q_i^*=mathbb E[M_iM_i^T],bbE[M_iM_i^T], mathsf K_k^*=0,sfK_k^*=0, mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_i(q_i^*-q_{i-1}^*), q_0^*=mathsf K_0^*b_0sfK_0^*b_0mathsf K_0^*.sfK_0^*.
-The inverse identity $mathsf K_isfK_imathsf P_isfP_imathsf K_i=sfK_i=mathsf K_i$sfK_i$
-on the active space, combined with the pinning across $theta$,
-identifies $q_0^*$ with the precision condition. The final statement
-is the canonical duality: the minimiser $C(b)$ of
-$Psi(C,b)-tr,C/2$ yields
+The inverse identity mathsf K_isfK_imathsf P_isfP_imathsf K_i=sfK_i=mathsf K_isfK_i
+on the active space, combined with the pinning across theta,
+identifies q_0^* with the precision condition. The final statement
+is the canonical duality: the minimiser C(b) of
+Psi(C,b)-tr,C/2 yields
-$$J(b):=Psi(C(b),b)-tfrac12tr(C(b)+b_{k-1})=S_theta(q^*)-frac12sum_iw_itr(q_i^*b_i)=inf_q{S_theta(q)-frac12sum_iw_itr(q_ib_i)},$$
+J(b):=Psi(C(b),b)-tfrac12tr(C(b)+b_{k-1})=S_theta(q^*)-frac12sum_iw_itr(q_i^*b_i)=inf_q{S_theta(q)-frac12sum_iw_itr(q_ib_i)},
-and $q^*$ is the unique minimiser of the right side. The path $q^*$ is
-eligible. For fixed $theta,d$ and bounded $b_{k-1}$, all such
+and q^* is the unique minimiser of the right side. The path q^* is
+eligible. For fixed theta,d and bounded b_{k-1}, all such
multipliers are bounded uniformly over the grids. The tilted terminal
field has uniformly bounded second and fourth moments.
Uniqueness and value identification. The entropy functional
-$S_theta$ is strictly convex in the path in the
-$2S=trPq+...$ sense: expanding $A_{i-1}=A_i+zeta_i(e_{i-1}-e_i)$
+S_theta is strictly convex in the path in the
+2S=trPq+... sense: expanding A_{i-1}=A_i+zeta_i(e_{i-1}-e_i)
successively cancels cross terms and leaves a sum of squares. In
-addition $S_theta(q^0)=0$ and the gradient at $q^0$ vanishes; hence
-$S_thetage 0$. At the canonical path, $mathsf P_0q_0^*sfP_0q_0^*mathsf P_0=b_0$.sfP_0=b_0$.
-Compress the inverse identity to each nested $F_i$ and telescope; the
+addition S_theta(q^0)=0 and the gradient at q^0 vanishes; hence
+S_thetage 0. At the canonical path, mathsf P_0q_0^*sfP_0q_0^*mathsf P_0=b_0.sfP_0=b_0.
+Compress the inverse identity to each nested F_i and telescope; the
scaled entropy gradient in the rightmost expression becomes
-$b_i|_{F_i}$. Uniqueness of the minimiser then holds.
+b_i|_{F_i}. Uniqueness of the minimiser then holds.
The dual values agree. Vary the field levels along the segment from
-zero to $b$. Covariance differentiation of $Psi$ before subtracting
-the diagonal shift inserts $tfrac12(mathsf K_i^*+sfK_i^*+zeta_imathbb
-EM_iM_i^T)$bbEM_iM_i^T)$ for an increment at level $i$. Telescoping and
-$mathbb Ess^T=I$bbEss^T=I$ give
+zero to b. Covariance differentiation of Psi before subtracting
+the diagonal shift inserts tfrac12(mathsf K_i^*+sfK_i^*+zeta_imathbb
+EM_iM_i^T)bbEM_iM_i^T) for an increment at level i. Telescoping and
+mathbb Ess^T=IbbEss^T=I give
-$$dot J=-tfrac12sum_iw_itr(q_i^*dot b_i),$$
+dot J=-tfrac12sum_iw_itr(q_i^*dot b_i),
and the entropy infimum has the same derivative by its unique
minimiser. Envelope differentiation is justified by the compact
-multiplier bound. Both values are zero at $b=0$; the minimiser there is
-$C=0$ and $q=q^0$.
+multiplier bound. Both values are zero at b=0; the minimiser there is
+C=0 and q=q^0.
Upper bound for the canonical integral. Restrict the Stiefel integral
to a small neighborhood of the identity in normalised triangular
lengths and cross readings. On that neighborhood the spin directions
have the exact Stiefel sharing law. Replacing the quadratic energy by
-$n,tr,C/2$ costs $o_ε(n)$, and replacing the
+n,tr,C/2 costs o_ε(n), and replacing the
linear-field covariance by that of the exact Stiefel columns also
-costs $o_ε(n)$: the normalised covariance discrepancy is
+costs o_ε(n): the normalised covariance discrepancy is
uniformly small, and Gaussian interpolation bounds the log change by
-half the diagonal-minus-pair trace difference. Letting $nto∞$
+half the diagonal-minus-pair trace difference. Letting nto∞
and then shrinking the neighborhood gives a lower bound on the
-canonical integral, so $limsup_nell_n(b)le J(b)$. The same
+canonical integral, so limsup_nell_n(b)le J(b). The same
covariance comparison and the compact multiplier bound give local
-uniformity in $b$.
+uniformity in b.
-Upper bound for the Gaussian theorem. Fix a trial path $q$ in the
-Gaussian functional $mathcal G_calG_{theta,p}$. The proof produces an
-expected pressure upper bound that matches $mathcal G_calG_{theta,p}$ at
-this trial, uniformly in $q$; taking the infimum over $q$ gives the
+Upper bound for the Gaussian theorem. Fix a trial path q in the
+Gaussian functional mathcal G_calG_{theta,p}. The proof produces an
+expected pressure upper bound that matches mathcal G_calG_{theta,p} at
+this trial, uniformly in q; taking the infimum over q gives the
theorem. The comparison follows Mourrat's supersolution method for
mean-field spin glasses with vector spins, adapted to the
nonlinear pattern Hamiltonian.
@@ -574,54 +574,54 @@ nonlinear pattern Hamiltonian.
./pdf/arxiv-1906.08471v2.pdf
Enrich the model. Add independent linear Gaussian fields with
-covariance levels tied to $q$. Replace the deterministic pattern
-count $M$ by Poisson$(Mr)$ with $rin[0,1]$, so that $r=0$ gives no
-patterns and $r=1$ recovers the true problem. Add independent
+covariance levels tied to q. Replace the deterministic pattern
+count M by Poisson(Mr) with rin[0,1], so that r=0 gives no
+patterns and r=1 recovers the true problem. Add independent
monomial perturbations of the pattern Hamiltonian scaled by
-$s_r=rsqrt M e_M$ with $e_M=M^{-1/16}$ and suitably normalised
+s_r=rsqrt M e_M with e_M=M^{-1/16} and suitably normalised
coefficients. The perturbation kernels are those scalarizations; the Gaussian realizations are achieved by tensor fields on
the spin columns.
Define the contact functional
-$$A_{M,r}(b,u)=-mathbb EbbEmathcal P_calP_{M,r}(b,u)+sum_jp_jS_theta(tildetilde q_q_{j})+frac r{2}sum_iw_itr(tildetilde q_ib_i)+q_ib_i)+Delta(r,u),$$
+A_{M,r}(b,u)=-mathbb EbbEmathcal P_calP_{M,r}(b,u)+sum_jp_jS_theta(tildetilde q_q_{j})+frac r{2}sum_iw_itr(tildetilde q_ib_i)+q_ib_i)+Delta(r,u),
-where $mathcal P_calP_{M,r}$ is the expected (perturbed, enriched)
-pressure, $tildetilde q_j$q_j$ is the covariance of the linear field, $b$ is
-the trial linear-field covariance path, $u$ collects the perturbation
-parameters, and $Delta$ is a small penalty that keeps the
-perturbation sizes and $b$ in a compact window.
+where mathcal P_calP_{M,r} is the expected (perturbed, enriched)
+pressure, tildetilde q_jq_j is the covariance of the linear field, b is
+the trial linear-field covariance path, u collects the perturbation
+parameters, and Delta is a small penalty that keeps the
+perturbation sizes and b in a compact window.
-Contact-point argument. Suppose the trial $q$ violates the desired
-bound, so that the true $r=1$ expected pressure exceeds
-$mathcal G_calG_{theta,p}$ at that trial by $3delta$ for some
-$delta>0$. Then $A_{M,0}(0,u)$ is $O(e_M^2)$ and $A_{M,1}(b_{r=1},u)$
+Contact-point argument. Suppose the trial q violates the desired
+bound, so that the true r=1 expected pressure exceeds
+mathcal G_calG_{theta,p} at that trial by 3delta for some
+delta>0. Then A_{M,0}(0,u) is O(e_M^2) and A_{M,1}(b_{r=1},u)
is negative. A large perturbation penalty is nonnegative. There is
-thus an interior minimiser of $A$ with $r>0$, small perturbation
+thus an interior minimiser of A with r>0, small perturbation
parameters, and the field cap not binding. At this minimiser the
first-order optimality conditions act as supersolution constraints.
-Differentiating in $r$ identifies the expected log-pressure of one
-fresh pattern with the negative of $r∂ A/∂ r$ at the
-minimiser. The latter is bounded below by $-V_D(q^*)-delta/2$, and
+Differentiating in r identifies the expected log-pressure of one
+fresh pattern with the negative of r∂ A/∂ r at the
+minimiser. The latter is bounded below by -V_D(q^*)-delta/2, and
this is the upper bound on the fresh pattern log-pressure.
-Differentiating in $b$ gives the tail contact inequality: for every
-nondecreasing variation $tildetilde bbsucceq b$ of the linear-field path
-with compact support, the covariance comparison $V_D(tildetilde q)q)le
-V_D(q)$ (from the path-order implication) implies
-$q^*preceq q$ in the Loewner order path sense. The gradient of the
-supersolution with respect to $b$ is a signed measure whose negative
-part is supported where $q^*$ stays below $q$. Summation by parts and
+Differentiating in b gives the tail contact inequality: for every
+nondecreasing variation tildetilde bbsucceq b of the linear-field path
+with compact support, the covariance comparison V_D(tildetilde q)q)le
+V_D(q) (from the path-order implication) implies
+q^*preceq q in the Loewner order path sense. The gradient of the
+supersolution with respect to b is a signed measure whose negative
+part is supported where q^* stays below q. Summation by parts and
the contact inequality at the minimiser identify the distribution of
-$q^*$ with a path of the same type.
+q^* with a path of the same type.
Ghirlanda-Guerra identities. The critical step is that, at the
minimising perturbation parameters, the joint Ghirlanda-Guerra
-identities hold in the limit $Mto∞$, and the matrix overlap
+identities hold in the limit Mto∞, and the matrix overlap
limits form an eligible matrix path. Supply of the GG identities uses
the monomial perturbations. Concentration of
-$M^{-1}logmathcal Z_calZ_{M,theta}$ on a bounded window of the
+M^{-1}logmathcal Z_calZ_{M,theta} on a bounded window of the
perturbation parameters, combined with Gaussian interpolation on the
perturbing fields, bounds the expected first derivative of the
perturbed pressure by the amplitude of the perturbation. Markov's
@@ -634,40 +634,40 @@ result identifies the matrix overlap of two replicas as a matrix
valued function of the label quantile, and the pinning and Loewner
order properties of the path quantile are exactly the properties
required for the field at the minimiser to match an eligible trial
-path. Uniqueness of the path identifes $tildetilde q$q$ with the trial
-$q$, so the contact-point upper bound matches $mathcal G_calG_{theta,p}$
-at the trial $q$. Because the trial was arbitrary, the upper bound
-follows uniformly over $q$ by the compactness of the trial paths.
+path. Uniqueness of the path identifes tildetilde qq with the trial
+q, so the contact-point upper bound matches mathcal G_calG_{theta,p}
+at the trial q. Because the trial was arbitrary, the upper bound
+follows uniformly over q by the compactness of the trial paths.
The trial class used at this point is restricted to paths whose
susceptibility is strictly positive on every nonempty active space;
this is the eligibility condition. If the infimum over all eligible
-$q$ is not attained, a limiting argument along an infimising sequence
+q is not attained, a limiting argument along an infimising sequence
uses local uniformity in the Stiefel restriction bound and in the
contact-point continuity. The expected pressure converges to the
infimum, uniformly in the pattern average.
The bound on the field cap can be chosen so as not to bind before
-using any overlap identities. For fixed $0<eta<1$ and trial $q$, the
-path $q'=q+eta(I-q)$ is eligible. The weighted difference against
+using any overlap identities. For fixed 0<eta<1 and trial q, the
+path q'=q+eta(I-q) is eligible. The weighted difference against
cumulative fields is
-$$sum_iw_itr((q_i'-q_i)b_i)=etasum_rtr(Delta b_rsum_{ige r}w_i(I-q_i))ge c_qsum_rtr,Delta b_r,$$
+sum_iw_itr((q_i'-q_i)b_i)=etasum_rtr(Delta b_rsum_{ige r}w_i(I-q_i))ge c_qsum_rtr,Delta b_r,
with
-$$c_q=etamin_{r:F_rne{0}}{w_rlambda_{min}((I-q_r)|_{F_r})}>0,$$
+c_q=etamin_{r:F_rne{0}}{w_rlambda_{min}((I-q_r)|_{F_r})}>0,
-where strict positivity holds because $I-q_rsucceqmathsf K_rsfK_rsucc 0$
-on $F_r$. Applying the Stiefel restriction bound with $q'$, bounded
-pattern energy, and the $O(e_M^2)$ perturbation cost, the contact
+where strict positivity holds because I-q_rsucceqmathsf K_rsfK_rsucc 0
+on F_r. Applying the Stiefel restriction bound with q', bounded
+pattern energy, and the O(e_M^2) perturbation cost, the contact
functional is positive at a sufficiently large fixed trace cap. Thus a
-negative minimising point has $r>0$ and permits small increases of
+negative minimising point has r>0 and permits small increases of
every allowed field increment.
-This closes the expected-pressure upper bound when $D$ is smooth with
-compact support. Bounded continuous $D$ follows by approximation:
-bounded Lipschitz functions agreeing with $D$ uniformly on compacts
+This closes the expected-pressure upper bound when D is smooth with
+compact support. Bounded continuous D follows by approximation:
+bounded Lipschitz functions agreeing with D uniformly on compacts
give the same limit. The negative-quadratic tail class is handled by the soft-penalty construction.
Lower bound: cavity with two restored patterns. The proof follows the
@@ -679,110 +679,110 @@ expressed through an optimization procedure for which
ultrametric/hierarchical structures form only a subset of the
variational class." A related critical-point method for vector spin
glasses with nonconvex interactions is due to Chen and Mourrat. Fix
-smooth compactly supported $D$. Replace the pattern count $M$ by
-$M+ell$ for a fixed cavity size $ell$ large compared to $1$ and tiny
-compared to $M$. Compare the enlarged pressure with the base $M$
+smooth compactly supported D. Replace the pattern count M by
+M+ell for a fixed cavity size ell large compared to 1 and tiny
+compared to M. Compare the enlarged pressure with the base M
pressure on the same disorder, using two restored patterns per
increment.
./pdf/aizenman-sims-starr-2003-cond-mat-0306386.pdf
./pdf/arxiv-2311.08980v3.pdf
-The cascade with only the outer split at $theta$ and the same
-monomial perturbations (amplitude $s_M=M^{3/8}$) replaces the
+The cascade with only the outer split at theta and the same
+monomial perturbations (amplitude s_M=M^{3/8}) replaces the
enriched upper-bound model. Average the expected logarithms over a
-product probability law $pi$ on the perturbation coefficients with
-each coefficient uniform on $[1,2]$, at every system size. The
-perturbation changes a log partition function by $O(s_M^2)=o(M)$.
+product probability law pi on the perturbation coefficients with
+each coefficient uniform on [1,2], at every system size. The
+perturbation changes a log partition function by O(s_M^2)=o(M).
Generic parameters. There is a set of perturbation parameters of
-$pi$-measure tending to one on which every subsequential overlap
+pi-measure tending to one on which every subsequential overlap
limit of the base system with two restored patterns satisfies joint
GG. Fix one perturbation coordinate. The pressure variance is
-$O(M^{-1})$; integration by parts bounds its expected first derivative
-by $O(s_M^2/M)$, and integrating the second derivative controls
+O(M^{-1}); integration by parts bounds its expected first derivative
+by O(s_M^2/M), and integrating the second derivative controls
thermal variance. Integrated convex difference quotients with step
-$s$ give
+s give
-$$C_a,s,s_M^2/M+C/(ssqrt M)$$
+C_a,s,s_M^2/M+C/(ssqrt M)
for the integrated absolute fluctuation of the first derivative. At
-$s=M^{-1/8}$ this implies
+s=M^{-1/8} this implies
-$$intmathbb E<|Y_a-bbE<|Y_a-mathbb E<bbE< Y_a>|>/s_M,dpito 0.$$
+intmathbb E<|Y_a-bbE<|Y_a-mathbb E<bbE< Y_a>|>/s_M,dpito 0.
The estimates hold on a slightly larger coupling interval so endpoint
difference quotients cause no difficulty, and a diagonal selection
over countably many coordinates gives the good sets.
-Restoring patterns. Fix a batch of $ell$ fresh independent patterns.
+Restoring patterns. Fix a batch of ell fresh independent patterns.
In the enlarged model, draw spin columns by Gram-Schmidt: the first
column is drawn at its prescribed vertex, and the remaining columns
are drawn at visits. Restrict all normalised triangular lengths and
cross readings to a fixed small neighbourhood of the identity. The
first restriction and each conditional inner restriction have
probability tending to one independently of the preceding frame, and
-the recursion normalisation costs are $o(n)$.
+the recursion normalisation costs are o(n).
-The enlarged Gibbs measure with the $M$ original patterns and the
-$ell$ fresh ones has self covariance close to the identity and cross
+The enlarged Gibbs measure with the M original patterns and the
+ell fresh ones has self covariance close to the identity and cross
covariance close to the empirical overlap of the sampled pair. This is
the physical prediction from spatial rotation invariance; it holds in
the enlarged Gibbs law for each fixed coordinate, uniformly over the
cavity coefficient, at expected probability tending to one.
-Thick shell. For fixed ratio $r_j$ of retained directions, the
-entries of $Z_j$ converge to independent standard Gaussians. For any
+Thick shell. For fixed ratio r_j of retained directions, the
+entries of Z_j converge to independent standard Gaussians. For any
converging bounded outer column the conditional inner statement holds
as well. Representing a uniform frame by orthonormalised standard
Gaussian columns proves the conditional version.
-Fix small $ε>0$ and restrict the enlarged partition function to
+Fix small ε>0 and restrict the enlarged partition function to
the thick shell
-$$mathcal I=calI={|Z_j^TZ_j/r_j-I|leεhbox{ for every }j}.$$
+mathcal I=calI={|Z_j^TZ_j/r_j-I|leεhbox{ for every }j}.
This only lowers its logarithm. On the shell, replace the larger
-perturbation by the base perturbation at the same $u$. For fixed
-$ell$, overlaps differ by $O_ell(M^{-1})$, and the summable kernel
-Lipschitz constants, the amplitude $s_M$, and the exponentially small
-coefficient tail together imply an $O_ell(M^{-1/4})+o(1)$ covariance
+perturbation by the base perturbation at the same u. For fixed
+ell, overlaps differ by O_ell(M^{-1}), and the summable kernel
+Lipschitz constants, the amplitude s_M, and the exponentially small
+coefficient tail together imply an O_ell(M^{-1/4})+o(1) covariance
error in unnormalised logarithms. Gaussian interpolation gives a
vanishing cost both for expected logs and for bounded replica tests.
On the shell, the physical cross covariance of two fresh patterns
matches the empirical matrix overlap of the enlarged system. The self
part of the shell makes the discarded Gibbs mass small for large
-$ell$ via Markov. Perturbation replacement therefore preserves the
+ell via Markov. Perturbation replacement therefore preserves the
same prediction with the base overlap on the right side.
Taylor expansion on the shell. Expansion of the fresh pattern row
fields around the base values gives a linear Gaussian field coupled
to each row of the fresh spin columns, of the form
-$n_j^{-1/2}sum_{rle M}h_j^r(D'_j(xi^r))^T$. The exact denominator
-$√(n_j+r_j)$ may be replaced by $√(n_j)$ because $r_j$ is
-fixed. The conditional cross covariance is therefore $B_j(X,widetilde
-X)$ in the limit.
+n_j^{-1/2}sum_{rle M}h_j^r(D'_j(xi^r))^T. The exact denominator
+√(n_j+r_j) may be replaced by √(n_j) because r_j is
+fixed. The conditional cross covariance is therefore B_j(X,widetilde
+X) in the limit.
The deterministic second-order correction is a one-path quadratic
-function $W(X,Z)$ with coefficients given by retained one-path
+function W(X,Z) with coefficients given by retained one-path
averages. On the shell,
-$$W(X,Z)=frac12sum_jr_jc_j(X)+O(εell).$$
+W(X,Z)=frac12sum_jr_jc_j(X)+O(εell).
-The first-order upper triangular part of $T_j-I$ has symmetric sum
-$-Z_j^TZ_j/n_j$; at $Z_j^TZ_j=r_jI$ this is $-r_jI/(2n_j)$. The mean
+The first-order upper triangular part of T_j-I has symmetric sum
+-Z_j^TZ_j/n_j; at Z_j^TZ_j=r_jI this is -r_jI/(2n_j). The mean
fresh quadratic term is the Hessian contracted with
-$Z_j^TZ_j/(2n_j)$. The diagonal specialisation is $r_jc_j(X)/2$,
+Z_j^TZ_j/(2n_j). The diagonal specialisation is r_jc_j(X)/2,
and the retained matrix coefficients are bounded, so moving within
-the shell costs $O(ε r_j)$. Summing over $j$ gives the
+the shell costs O(ε r_j). Summing over j gives the
correction. Cross-species fresh quadratic terms are centered.
Canonical prediction. Restricted to the thick shell, the Gaussian
predicts one specific matrix overlap path for the fresh-coordinate
pattern pair. Conditioning on the transformed unmarked sampled shape,
the canonical spatial rows are iid jointly Gaussian across replicas;
-the self covariance is $I$ and the cross covariance at common
-level $i$ is $q_{j,i}^*$ from the canonical lemma. The independent root
+the self covariance is I and the cross covariance at common
+level i is q_{j,i}^* from the canonical lemma. The independent root
rows are averaged in this assertion, and additive row energies with
new-pattern energies factor the tilted kernels, so new patterns do not
bias these row laws.
@@ -790,213 +790,213 @@ bias these row laws.
The comparison. Physical rotation invariance predicts that the
sampled cross overlap approximates the empirical matrix overlap of the
enlarged system. Gaussian calculation on the same thick shell
-predicts that the same sampled cross overlap approximates $q_{j,i}^*$.
+predicts that the same sampled cross overlap approximates q_{j,i}^*.
Both predictions hold at the sample coordinate level, and the
failure probabilities bound the weighted path distance by twice the
tolerance plus a bounded multiple of the sum. Therefore the
empirical matrix overlap of the enlarged system is close to the
-canonical path $q^*$.
+canonical path q^*.
-Summing the pieces. Let $V_D(q)$ denote the limiting expected log
-increment of one independent pattern at a GG limit; the entropy $S_theta$
-and the Gaussian functional $mathcal G_calG_{theta,p}$ already entered
+Summing the pieces. Let V_D(q) denote the limiting expected log
+increment of one independent pattern at a GG limit; the entropy S_theta
+and the Gaussian functional mathcal G_calG_{theta,p} already entered
via the canonical lemma. The cavity comparison identifies the
-enlarged pressure with $ell$ times the canonical value minus $o(ell)$
+enlarged pressure with ell times the canonical value minus o(ell)
error, so
-$$frac1ellmathbb E[bbE[logmathcal Z_calZ_{M+ell}-logmathcal Z_M]calZ_M]gemathcal G_calG_{theta,p}(D)-o_{tol}(1).$$
+frac1ellmathbb E[bbE[logmathcal Z_calZ_{M+ell}-logmathcal Z_M]calZ_M]gemathcal G_calG_{theta,p}(D)-o_{tol}(1).
-Uniformity in the base size. For every $delta>0$ there is $ell_0$
-such that for each integer $ellgeell_0$, every sequence of base
-sizes with $n_j/Mto p_j$ and increments $r_j=p_jell+O(1)$ satisfies
+Uniformity in the base size. For every delta>0 there is ell_0
+such that for each integer ellgeell_0, every sequence of base
+sizes with n_j/Mto p_j and increments r_j=p_jell+O(1) satisfies
-$$liminf_{Mto∞}frac1ellintDelta_{M,ell}(u),dpi(u)gemathcal G_calG_{theta,p}(D)-delta,$$
+liminf_{Mto∞}frac1ellintDelta_{M,ell}(u),dpi(u)gemathcal G_calG_{theta,p}(D)-delta,
-where $Delta_{M,ell}$ is the log-increment of the enlarged pressure
-beyond the base at parameter $u$. The constant in $O(1)$ is fixed
+where Delta_{M,ell} is the log-increment of the enlarged pressure
+beyond the base at parameter u. The constant in O(1) is fixed
across these choices. Begin with any good-parameter sequence and any
compact array limit. The tolerance, shell, and rounding choices apply
to it with uniform canonical constants; failure of the eventual bound
would contradict the cavity inequality. The bad parameter sets
-have $pi$-measure zero and use the fixed-$ell$ uniform lower
+have pi-measure zero and use the fixed-ell uniform lower
increment bound.
-Telescoping. Along the triangular chain $M'$ with $M'equiv M$
-modulo $ell$ and block dimensions $lfloor M'n_j/Mrfloor$, divide by
-$M'$ and send $M'to∞$, then $ellto∞$ and $delta↓ 0$.
-The same parameter law $pi$ at both endpoints makes log increments
-telescope exactly; the $o(M)$ perturbation error is removable. This
-proves the matching lower bound for smooth compactly supported $D$.
+Telescoping. Along the triangular chain M' with M'equiv M
+modulo ell and block dimensions lfloor M'n_j/Mrfloor, divide by
+M' and send M'to∞, then ellto∞ and delta↓ 0.
+The same parameter law pi at both endpoints makes log increments
+telescope exactly; the o(M) perturbation error is removable. This
+proves the matching lower bound for smooth compactly supported D.
The bounded continuous case follows by approximation; the
convolution of the enlargement with the compact approximation equals
-the pressure at the larger argument up to $o(M)$, uniformly on shells.
+the pressure at the larger argument up to o(M), uniformly on shells.
Maximal sub-sequential identification. Combining the upper and lower
bounds, every subsequential limit of the normalised expected log
-pressure equals $mathcal G_calG_{theta,p}(D)$. Concentration of the
+pressure equals mathcal G_calG_{theta,p}(D). Concentration of the
normalised log on a bounded window transfers this
-to convergence in probability. Boundedness of $D$ and uniform
+to convergence in probability. Boundedness of D and uniform
integrability bound the pressure, giving convergence in expectation.
-Soft penalties and quadratic tails. The Gaussian theorem was proved for smooth compactly supported $D$,
+Soft penalties and quadratic tails. The Gaussian theorem was proved for smooth compactly supported D,
and the bounded continuous case follows by approximation. The angular formula
-also requires a negative-quadratic tail class. Extend $D$ to be the
+also requires a negative-quadratic tail class. Extend D to be the
sum of a bounded continuous function and a nonpositive quadratic form
-$-C|xi|^2$. Uniform tail control for Gaussian rows under the trial
+-C|xi|^2. Uniform tail control for Gaussian rows under the trial
path is separate from the approximation, and both are needed.
-Row tails. For every trial path $q$ and every $R<∞$ the
-Gaussian row fields satisfy $mathbb EbbEexp{c|Z_j|^2}le C_{c,R}$ for
-$|c|$ small enough and all rows $rle M$; the constants are uniform in
+Row tails. For every trial path q and every R<∞ the
+Gaussian row fields satisfy mathbb EbbEexp{c|Z_j|^2}le C_{c,R} for
+|c| small enough and all rows rle M; the constants are uniform in
the finite grid and the trial path. Bounded smooth approximation of
-$D$ on compacts, together with a Markov tail bound on the residual
-exponential mass, gives an $L^1$ difference of the normalised log
+D on compacts, together with a Markov tail bound on the residual
+exponential mass, gives an L^1 difference of the normalised log
partition functions tending to zero. The same comparison applies to
-$V_D$ by the terminal class' explicit recursion.
+V_D by the terminal class' explicit recursion.
Distinguished negative-quadratic. The soft-penalty Hamiltonian
-$H=sum_rD(xi_r)-Mdsum_ell(barbar g_g_ell-c_ell)_+$ is exactly the
+H=sum_rD(xi_r)-Mdsum_ell(barbar g_g_ell-c_ell)_+ is exactly the
shape in which a nonpositive quadratic enters. Rewrite the soft
-penalty using the convex dual of $x_+$:
+penalty using the convex dual of x_+:
-$$-Mdsum_ell(barbar g_g_ell-c_ell)_+=min_{0le t_ellle d}M(tcdot c-sum_ell t_ellbarbar g_g_ell).$$
+-Mdsum_ell(barbar g_g_ell-c_ell)_+=min_{0le t_ellle d}M(tcdot c-sum_ell t_ellbarbar g_g_ell).
The minimum is attained because the constraint set is compact and the
-objective is linear in $t$. Substituting yields a Hamiltonian of the
-form $H=sum_r(D(xi_r)-tcdot g(xi_r))$ with bounded linear-$t$
+objective is linear in t. Substituting yields a Hamiltonian of the
+form H=sum_r(D(xi_r)-tcdot g(xi_r)) with bounded linear-t
coefficients. The unpenalised Gaussian theorem with terminal
-$D-tcdot g$ applies uniformly in $tin[0,d]^s$ because negative
+D-tcdot g applies uniformly in tin[0,d]^s because negative
quadratic coefficients remain negative, so it covers every
-$t$-terminal in one stroke.
+t-terminal in one stroke.
Concentration of the average. Under the natural tilted path law at
-parameter $t$, the empirical average $barbar g$g$ of the soft-constraint
+parameter t, the empirical average barbar gg of the soft-constraint
functions concentrates in probability at a deterministic vector
-$v(t)$, obtained as a limit of $barbar g$g$ along compactly supported
-clippings. For each direction $a$ into the compact box $[0,d]^s$, the
-one-sided derivative of the pressure with respect to $t$ is
+v(t), obtained as a limit of barbar gg along compactly supported
+clippings. For each direction a into the compact box [0,d]^s, the
+one-sided derivative of the pressure with respect to t is
-$$∂_ap(t)=-acdot v(t).$$
+∂_ap(t)=-acdot v(t).
Passage of slopes through clipping: interpolate only the increment in
the coupling. Increasing a nonnegative quadratic coupling adds a
-nonpositive term; decreasing along $a_ell<0$ and $u>0$ contributes,
-for a bounded clipping $g_{ell,R}$,
+nonpositive term; decreasing along a_ell<0 and u>0 contributes,
+for a bounded clipping g_{ell,R},
-$$-(t_ell-u|a_ell|)g_ell-u|a_ell|(1-gamma)(g_ell-g_{ell,R}),qquad 0le gammale 1,$$
+-(t_ell-u|a_ell|)g_ell-u|a_ell|(1-gamma)(g_ell-g_{ell,R}), 0le gammale 1,
with nonpositive unbounded coefficients on both terms. Uniform tails
from the row-tail estimate make the directional slope error at most
-$sum_ell|a_ell|ε_R$, with $ε_Rto 0$. This proves the
+sum_ell|a_ell|ε_R, with ε_Rto 0. This proves the
one-sided derivative formula, including at boundary directions.
Duality in closed form. Once the empirical average concentrates and
the derivative formula holds, standard convex duality in finite
dimensions gives the soft-penalty pressure as
-$$min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.$$
+min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.
-The unpenalised functional $mathcal G_calG_{theta,p}(D-tcdot g)$ is
-jointly convex in $t$ by direct differentiation and continuity of the
-unpenalised pressure in $h$ under the terminal class; no separate
+The unpenalised functional mathcal G_calG_{theta,p}(D-tcdot g) is
+jointly convex in t by direct differentiation and continuity of the
+unpenalised pressure in h under the terminal class; no separate
lower-semicontinuity hypothesis is needed, because the compact box and
-boundedness of $D$ on compacts ensure it. The minimum is attained, and
-the value is finite and continuous in $(c,D,g)$ on compacts.
+boundedness of D on compacts ensure it. The minimum is attained, and
+the value is finite and continuous in (c,D,g) on compacts.
Uniform tail estimate. The same extension covers the negative
-quadratic. Write $D=D_0-C|xi|^2$ with $D_0$ bounded continuous. On a
+quadratic. Write D=D_0-C|xi|^2 with D_0 bounded continuous. On a
good spectral event of probability tending to one, every input has
-$(Ax)$ of squared norm bounded above by $(N/M)C'^2$; on the same event
-bounded smooth approximations of $D$ on compacts and the negative
-quadratic agree with $D$ outside a set of exponentially small field
-mass. The $L^1$ difference of log partition functions is $o(M)$
+(Ax) of squared norm bounded above by (N/M)C'^2; on the same event
+bounded smooth approximations of D on compacts and the negative
+quadratic agree with D outside a set of exponentially small field
+mass. The L^1 difference of log partition functions is o(M)
uniformly on trials. This closes the Gaussian theorem for terminal
-class $D=D_0-C|xi|^2$.
+class D=D_0-C|xi|^2.
-Haar reduction to Gaussian fields. Let $G$ be an $Mtimes n$ matrix
-with independent $N(0,1/M)$ entries, $n=sum_jn_j$. Gaussian
+Haar reduction to Gaussian fields. Let G be an Mtimes n matrix
+with independent N(0,1/M) entries, n=sum_jn_j. Gaussian
orthogonalization of its consecutive column blocks of sizes
-$n_1,n_2,...$ gives a Haar flag
-$E_1subset E_1 ⊕ E_2subset...$ of the desired dimensions.
-On an event of probability at least $1-e^{-cM}$, the operator norm of
-$G$ and of its inverse on its image are bounded by fixed constants
-$C_{op},C_{inv}$. An elementary net argument gives
-the bound. An $r$-dimensional unit sphere has a $delta$-net of size
-at most $(1+2/delta)^r$; fixed-vector Gaussian norms have arbitrarily
+n_1,n_2,... gives a Haar flag
+E_1subset E_1 ⊕ E_2subset... of the desired dimensions.
+On an event of probability at least 1-e^{-cM}, the operator norm of
+G and of its inverse on its image are bounded by fixed constants
+C_{op},C_{inv}. An elementary net argument gives
+the bound. An r-dimensional unit sphere has a delta-net of size
+at most (1+2/delta)^r; fixed-vector Gaussian norms have arbitrarily
strong exponential upper tails at a large constant, with small-ball
-probability at most $(Cε)^M$. Take a fixed mesh for the upper
-bound and a mesh of order $ε$ for the lower bound; the strict
-inequality $n/Mto p<1$ makes the small-ball exponent dominate the
+probability at most (Cε)^M. Take a fixed mesh for the upper
+bound and a mesh of order ε for the lower bound; the strict
+inequality n/Mto p<1 makes the small-ball exponent dominate the
net size.
./pdf/arxiv-mezzadri-0609050.pdf
-The volume Jacobian of $xmapsto Gx$ on the coefficient sphere is
-$√(det(G^TG))$. Its logarithm divided by $M$ converges in
+The volume Jacobian of xmapsto Gx on the coefficient sphere is
+√(det(G^TG)). Its logarithm divided by M converges in
probability to
-$$J_p=frac12int_0^plog(1-u),du.$$
+J_p=frac12int_0^plog(1-u),du.
-Indeed the successive orthogonal residual lengths of $G$ are
-independent with laws $(chi^2_{M-r+1}/M)^{1/2}$, chi-square
+Indeed the successive orthogonal residual lengths of G are
+independent with laws (chi^2_{M-r+1}/M)^{1/2}, chi-square
concentration bounds the normalised logarithms, and a Riemann sum
gives the integral.
-Coefficient integral. Write $A=Udiag(s)V^T$ and put
-$x=Vw$ so that $x$ is uniform on $S^N$. The column blocks of $A$
-multiply $x$ block by block: $Ax=sum_jU_jdiag(s_j)V_j^Tx$.
-On the Haar flag the blocks $V_j^Tx$ of a uniform coefficient decompose
-as independent uniform directions in $mathbb R^bbR^{n_j}$ up to the
+Coefficient integral. Write A=Udiag(s)V^T and put
+x=Vw so that x is uniform on S^N. The column blocks of A
+multiply x block by block: Ax=sum_jU_jdiag(s_j)V_j^Tx.
+On the Haar flag the blocks V_j^Tx of a uniform coefficient decompose
+as independent uniform directions in mathbb R^bbR^{n_j} up to the
Jacobian. Thus the angular pressure satisfies
-$$P_M^f(a)=frac1Mlogint_{S^n}e^{sum_rf((Gx)_r)}|det(G^TG)|^{1/2}dsigma_n(x)+o(1).$$
+P_M^f(a)=frac1Mlogint_{S^n}e^{sum_rf((Gx)_r)}|det(G^TG)|^{1/2}dsigma_n(x)+o(1).
The change of measure is exact: it transfers the Haar flag weight to
-the Gaussian matrix $G$ with its Jacobian. The error $o(1)$ comes from
+the Gaussian matrix G with its Jacobian. The error o(1) comes from
a bounded activation and the good-norm event.
-Witnesses. Introduce auxiliary unit-block vectors $z_{i,j}inmathbb
-R^bbR^{n_j}$ for $jle i<h$ and put
+Witnesses. Introduce auxiliary unit-block vectors z_{i,j}inmathbb
+R^bbR^{n_j} for jle i<h and put
-$$u_i=sum_{jle i}G_jz_{i,j}.$$
+u_i=sum_{jle i}G_jz_{i,j}.
-The test $|y-u_i|^2/Mlesum_{j>i}a_j$ forces the squared norm in
-each tail of the orthogonal decomposition of $y$ to be no larger than
+The test |y-u_i|^2/Mlesum_{j>i}a_j forces the squared norm in
+each tail of the orthogonal decomposition of y to be no larger than
the prescribed tail, so that each initial block carries at least its
-prescribed total mass. After sorting $a_j/p_j$, the weighted shell
+prescribed total mass. After sorting a_j/p_j, the weighted shell
lemma shows that these one-sided tests recover the desired weighted
-shell of radii $a_j$. The witnesses are integrated as the inner
-variables of the powered Stiefel integral, with exponent $thetato 0$
+shell of radii a_j. The witnesses are integrated as the inner
+variables of the powered Stiefel integral, with exponent thetato 0
removing the entropy.
-Coefficient model. With bounded $f$, the coefficient integral over
-$S^n$ decomposes into $intprod_jdmu_j$ where $mu_j$ is uniform on
-$S^{n_j}$, and the Gaussian column blocks $G_j$ act on each $x_j$
-independently. Encoding the constraint $|y-u_i|^2/Mle
-sum_{j>i}a_j$ via the soft penalty at multiplier $t_ell$ and sending
-the cap $lambda/theta$ to infinity gives exactly the functional
-$mathcal B_calB_{p,a}(kappa,lambda,B,theta)$ after taking a supremum
-over the trial coefficient frames $R_j$ that parametrise the inner
+Coefficient model. With bounded f, the coefficient integral over
+S^n decomposes into intprod_jdmu_j where mu_j is uniform on
+S^{n_j}, and the Gaussian column blocks G_j act on each x_j
+independently. Encoding the constraint |y-u_i|^2/Mle
+sum_{j>i}a_j via the soft penalty at multiplier t_ell and sending
+the cap lambda/theta to infinity gives exactly the functional
+mathcal B_calB_{p,a}(kappa,lambda,B,theta) after taking a supremum
+over the trial coefficient frames R_j that parametrise the inner
witness block-by-block.
-The weighted shell lemma. Sort the numbers $a_j/p_j$ in nonincreasing
-order, and suppose $b_jge 0$ satisfy
+The weighted shell lemma. Sort the numbers a_j/p_j in nonincreasing
+order, and suppose b_jge 0 satisfy
-$$sum_jb_j=sum_ja_j,qquad sum_{jle i}b_jgesum_{jle i}a_jquad(1le i<h).$$
+sum_jb_j=sum_ja_j, sum_{jle i}b_jgesum_{jle i}a_jquad(1le i<h).
Let
-$$Phi(a)=H_p(a)+P_∞(a),qquad H_p(a)=frac12sum_jp_jlog(2pi e a_j/p_j),$$
+Phi(a)=H_p(a)+P_∞(a), H_p(a)=frac12sum_jp_jlog(2pi e a_j/p_j),
-with $P_∞(a)=lim_Mprob-P_M^f(a)$ the deterministic
-angular limit. Then $Phi(b)lePhi(a)$.
+with P_∞(a)=lim_Mprob-P_M^f(a) the deterministic
+angular limit. Then Phi(b)lePhi(a).
-Proof. Put $A$ equal to the decreasing density with values $a_j/p_j$
-on the successive intervals of lengths $p_j$, and let $B^*$ equal the
-decreasing rearrangement of $b_j/p_j$ on the same lengths of intervals.
-At each original cutpoint, the initial integral of $B^*$ dominates the
-original $b$-prefix and hence the $a$-prefix. Between cutpoints the
+Proof. Put A equal to the decreasing density with values a_j/p_j
+on the successive intervals of lengths p_j, and let B^* equal the
+decreasing rearrangement of b_j/p_j on the same lengths of intervals.
+At each original cutpoint, the initial integral of B^* dominates the
+original b-prefix and hence the a-prefix. Between cutpoints the
former cumulative integral is concave and the latter affine. Thus
-initial integrals of $B^*$ dominate those of $A$ everywhere, with equal
+initial integrals of B^* dominate those of A everywhere, with equal
totals.
The allocation step is the finite-weight Hardy-Littlewood-Polya
@@ -1004,174 +1004,174 @@ convex-order principle. For the underlying rearrangement theory see
Hardy, Littlewood, and Polya's monograph; the specific weight-preserving
allocation proof is given.
-Consider allocation tables $lambda_{ij}ge 0$ with column sums $p_j$
-and row sums $p_i$. The entry $lambda_{ij}$ is the dimension
-proportion sent from old block $j$ to new block $i$; its squared-mass
-contribution is $lambda_{ij}b_j/p_j$. To show that some table attains
-$a$, fix arbitrary prices $c_i$, and let $C$ equal $c_i$ on the
-interval of length $p_i$. Write $C^*$ for its decreasing rearrangement.
+Consider allocation tables lambda_{ij}ge 0 with column sums p_j
+and row sums p_i. The entry lambda_{ij} is the dimension
+proportion sent from old block j to new block i; its squared-mass
+contribution is lambda_{ij}b_j/p_j. To show that some table attains
+a, fix arbitrary prices c_i, and let C equal c_i on the
+interval of length p_i. Write C^* for its decreasing rearrangement.
The support function of the attainable mass vectors assigns the largest
source densities to the largest prices. Rearrangement and then
cumulative dominance give
-$$sum_ic_ia_i=int_0^pCAleint_0^pC^*Aleint_0^pC^*B^*=max_lambdasum_{i,j}c_ilambda_{ij}b_j/p_j.$$
+sum_ic_ia_i=int_0^pCAleint_0^pC^*Aleint_0^pC^*B^*=max_lambdasum_{i,j}c_ilambda_{ij}b_j/p_j.
The second inequality is summation by parts; equal total mass permits
arbitrary, including negative, prices. Finite-dimensional separation
-gives a table attaining $a$. Set $T_{ij}=lambda_{ij}/p_j$; then
+gives a table attaining a. Set T_{ij}=lambda_{ij}/p_j; then
-$$Tge 0,qquad sum_iT_{ij}=1,qquad Tp=p,qquad Tb=a.$$
+Tge 0, sum_iT_{ij}=1, Tp=p, Tb=a.
-The relation is majorization relative to the block proportions $p$;
-see the class of $d$-majorization polytopes studied by vom Ende and
-Dirr. Realise it geometrically by randomly subdividing each old $E_j$
-into orthogonal pieces of relative dimensions approaching $T_{ij}$,
-and forming new block $i$ as the direct sum. The identities $Tp=p$
-and $Tb=a$ give respectively the new dimension proportions and the
+The relation is majorization relative to the block proportions p;
+see the class of d-majorization polytopes studied by vom Ende and
+Dirr. Realise it geometrically by randomly subdividing each old E_j
+into orthogonal pieces of relative dimensions approaching T_{ij},
+and forming new block i as the direct sum. The identities Tp=p
+and Tb=a give respectively the new dimension proportions and the
target squared masses. Dimensions with the correct two margins are
-obtained by $o(M)$ adjustments and bounded rounding corrections. The
+obtained by o(M) adjustments and bounded rounding corrections. The
new decomposition has the same Haar marginal as the old one. For each
-fixed vector in the old $b$-shell, squared norms in the pieces
+fixed vector in the old b-shell, squared norms in the pieces
concentrate about the prescribed fractions; the failure bound is
uniform in the fixed vector. Simultaneous concentration for all shell
vectors is unnecessary. Thus each such vector lies in a slightly
-enlarged new $a$-shell with failure probability $ε_Mto 0$.
+enlarged new a-shell with failure probability ε_Mto 0.
./pdf/arxiv-1911.01061v5.pdf
Angular identity. Radial integration and Stirling's formula show that
-the weighted Lebesgue volume in $E_1 ⊕ ... ⊕ E_h$ of a small
-shell with $|y_j|^2/M$ near a positive $a_j$ has logarithm per $M$,
-after shrinking the shell, equal to $Phi(a)=H_p(a)+P_∞(a)$. The
+the weighted Lebesgue volume in E_1 ⊕ ... ⊕ E_h of a small
+shell with |y_j|^2/M near a positive a_j has logarithm per M,
+after shrinking the shell, equal to Phi(a)=H_p(a)+P_∞(a). The
same radial calculation gives the upper Laplace bound on compact sets.
-Set $Phi=-∞$ at a zero coordinate; the divergent radial entropy
-and the bounded pressure $P_∞$ justify this convention in upper
+Set Phi=-∞ at a zero coordinate; the divergent radial entropy
+and the bounded pressure P_∞ justify this convention in upper
bounds.
The weighted shell lemma plus the witness construction completes the
angular reduction. The upper bound uses the shell volume estimate and
Laplace's method. The lower bound uses the transport realisation: a
-typical vector of the $b$-shell lies in the $a$-shell up to
-$ε$, so the integral on the $a$-shell is at least the integral
-on the $b$-shell up to $O(ε)$.
+typical vector of the b-shell lies in the a-shell up to
+ε, so the integral on the a-shell is at least the integral
+on the b-shell up to O(ε).
-Completing the angular formula. Fix $kappa$ from the working lower
-bound. Then fix any finite $lambda$ and a desired error $ε$.
+Completing the angular formula. Fix kappa from the working lower
+bound. Then fix any finite lambda and a desired error ε.
The concentration rate for the coefficient integral at that error is
-independent of $B,theta$. Choose $eta$ so that the entropy
-$frac12logeta$ is below half this rate and so that
-$2|f|_∞eta+frac12logeta<ε$. Choose shell widths and
-$delta$ so the errors times $lambda$ are small, then choose a
-sufficiently large $B$, independently of $theta$. Finally take
-$theta$ small enough that the witness entropy is below $ε$. A
+independent of B,theta. Choose eta so that the entropy
+frac12logeta is below half this rate and so that
+2|f|_∞eta+frac12logeta<ε. Choose shell widths and
+delta so the errors times lambda are small, then choose a
+sufficiently large B, independently of theta. Finally take
+theta small enough that the witness entropy is below ε. A
union bound over the candidate rotations and the concentration
estimate turns the witness lower bound into the corresponding
-deterministic bound for $J_p+mathcal B_calB_{p,a}$ with one additional
-$ε$ loss. Thresholds may depend on the fixed $lambda$; no
+deterministic bound for J_p+mathcal B_calB_{p,a} with one additional
+ε loss. Thresholds may depend on the fixed lambda; no
uniformity in its later limit is required.
-Monotonicities. For fixed $M$, the soft integral increases as
-$theta↓ 0$ because the inner expression is an increasing
-$L^{1/theta}$ norm. It increases with $B$ and decreases with
-$lambda$. These monotonicities pass to the deterministic limits. The
+Monotonicities. For fixed M, the soft integral increases as
+theta↓ 0 because the inner expression is an increasing
+L^{1/theta} norm. It increases with B and decreases with
+lambda. These monotonicities pass to the deterministic limits. The
upper bound and the witness lower bound establish the successive
limits and
-$$Phi(a)=J_p+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta).$$
+Phi(a)=J_p+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta).
-Using the shell-volume identity for $Phi$ proves the angular formula
-and identifies every subsequential $P_∞$. Equicontinuity extends
+Using the shell-volume identity for Phi proves the angular formula
+and identifies every subsequential P_∞. Equicontinuity extends
convergence to compact sets and to zero radii.
-Full output dimension. If $sum_jp_j=1$, retain, independently in each
-block, a random subspace of dimension $(1-eta)n_j+o(M)$. Under the
+Full output dimension. If sum_jp_j=1, retain, independently in each
+block, a random subspace of dimension (1-eta)n_j+o(M). Under the
untilted angular law, the fraction of squared norm lost in each block
-is at most $2eta$ with probability tending to one. The same is true of
+is at most 2eta with probability tending to one. The same is true of
the old Gibbs mass with probability tending to one over the chosen
retained subspaces: for each fixed old vector the projection failure
probability is uniformly small, so Fubini and Markov apply after
arbitrary bounded activation reweighting. Zero-radius blocks are
-ignored. Letting $eta↓ 0$ recovers the $p=1$ case.
+ignored. Letting eta↓ 0 recovers the p=1 case.
-Continuity in $f$ and $a$. For $fin C_b(mathbb R)$bbR)$ choose bounded
+Continuity in f and a. For fin C_b(mathbb R)bbR) choose bounded
Lipschitz approximants with a common bound and uniform convergence on
compact intervals. Every angular configuration satisfies
-$M^{-1}sum_ry_r^2=sum_ja_j$. On a compact radius set the fraction of
-entries outside $[-R,R]$ is uniformly $O(R^{-2})$. Compact
+M^{-1}sum_ry_r^2=sum_ja_j. On a compact radius set the fraction of
+entries outside [-R,R] is uniformly O(R^{-2}). Compact
approximation controls the normalised energy uniformly and transfers
all angular conclusions. The same estimate applies to
-$mathcal B_calB_{p,a}$ on a bounded operator-norm event, independently of
-$lambda,B,theta$, so its limiting formula has the same continuity.
+mathcal B_calB_{p,a} on a bounded operator-norm event, independently of
+lambda,B,theta, so its limiting formula has the same continuity.
This completes the angular theorem.
-The radial decomposition. Return to the spherical integral with $A$ of
+The radial decomposition. Return to the spherical integral with A of
bi-orthogonally invariant law. Fix the singular-value block
-approximation $s_1,...,s_h$ with multiplicities $n_j$, and let $G$
+approximation s_1,...,s_h with multiplicities n_j, and let G
be the Gaussian flag matrix from the angular reduction. The
-decomposition of $A$ yields
+decomposition of A yields
-$$A=U,G,V^T$$
+A=U,G,V^T
-in law, with $U,V$ independent uniform orthogonal matrices, and with
-$G$ acting block by block on the coefficient $x=Vw$. The spherical
-coefficient $x$ decomposes into blocks of sizes $n_1,...,n_h$.
+in law, with U,V independent uniform orthogonal matrices, and with
+G acting block by block on the coefficient x=Vw. The spherical
+coefficient x decomposes into blocks of sizes n_1,...,n_h.
Write
-$$x=√(rho_0N)theta_0+sum_j√(rho_jN/s_j^2)theta_j√(s_j^2/N),$$
+x=√(rho_0N)theta_0+sum_j√(rho_jN/s_j^2)theta_j√(s_j^2/N),
-where $rho_jge 0$ is the fraction of squared norm in block $j$ (with a
-null block of size $N-M$ when $alpha<1$), $sum_jrho_j=1$, and the
-$theta_j$ are unit vectors in the blocks. For a uniform coefficient
-$x$ the joint law factors: the $theta_j$ are independent uniform unit
-vectors in the blocks, and $rho=(rho_j)$ is Dirichlet with
-parameters $(n_j/2)$ (including the null block when it is present).
+where rho_jge 0 is the fraction of squared norm in block j (with a
+null block of size N-M when alpha<1), sum_jrho_j=1, and the
+theta_j are unit vectors in the blocks. For a uniform coefficient
+x the joint law factors: the theta_j are independent uniform unit
+vectors in the blocks, and rho=(rho_j) is Dirichlet with
+parameters (n_j/2) (including the null block when it is present).
Under this split, the measurement is
-$Ax=sum_jU_j(s_j√(Nrho_j/alpha)),G_jtheta_j$ plus a null
-contribution. Block $j$ has image subspace $E_j=U_jmathbb R^bbR^{n_j}$ of
-Haar dimension $n_j$ in $mathbb R^M$,bbR^M$, and the vector
-$y_j=s_j√(Nrho_j)G_jtheta_j$ is uniform on the sphere of radius
-$√(Ma_j)$ in $E_j$ with $a_j=s_j^2rho_j/alpha$. The $y_j$ are
-mutually independent once $rho$ is fixed.
+Ax=sum_jU_j(s_j√(Nrho_j/alpha)),G_jtheta_j plus a null
+contribution. Block j has image subspace E_j=U_jmathbb R^bbR^{n_j} of
+Haar dimension n_j in mathbb R^M,bbR^M, and the vector
+y_j=s_j√(Nrho_j)G_jtheta_j is uniform on the sphere of radius
+√(Ma_j) in E_j with a_j=s_j^2rho_j/alpha. The y_j are
+mutually independent once rho is fixed.
Pressure as a Dirichlet supremum. Substitute into the pressure and use
the independence. On the coefficient sphere the factor
-$|det(G^TG)|^{1/2}$ contributes the Jacobian rate $J_p$. The
+|det(G^TG)|^{1/2} contributes the Jacobian rate J_p. The
normalised log of the integral over shell radii equals
-$$sup_{substack{rho_i>0sum_{iinmathcal I}calIrho_i=1}}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)}+o(1),$$
+sup_{substack{rho_i>0sum_{iinmathcal I}calIrho_i=1}}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)}+o(1),
-where $v_j=alpha p_j$, $v_0=1-alpha$ when $alpha<1$, and
-$mathcal I=calI={1,...,h}$ when $alphage 1$ and
-$mathcal I=calI={0,1,...,h}$ when $alpha<1$. The first term is the
+where v_j=alpha p_j, v_0=1-alpha when alpha<1, and
+mathcal I=calI={1,...,h} when alphage 1 and
+mathcal I=calI={0,1,...,h} when alpha<1. The first term is the
Dirichlet entropy of the split of squared norm, whose density near
-$rho$ is $rho_1^{n_1/2-1}...$, contributing
-$frac12sum_jp_jlog(rho_j/p_j)$ plus a normalisation giving
-$frac12v_jlog(rho_j/v_j)$ after the substitution. The second term is
-the angular pressure times $alpha$, converting per-$M$ pressure to
-per-$N$ pressure.
+rho is rho_1^{n_1/2-1}..., contributing
+frac12sum_jp_jlog(rho_j/p_j) plus a normalisation giving
+frac12v_jlog(rho_j/v_j) after the substitution. The second term is
+the angular pressure times alpha, converting per-M pressure to
+per-N pressure.
The supremum is attained, continuous, and bounded, because the
Dirichlet entropy is strictly concave on the simplex and the angular
pressure is bounded continuous by the angular theorem. The supremum's
-value depends continuously on the vector $s=(s_1,...,s_h)$ and on
-the multiplicities $p_j$, so the finite-block approximation is well
+value depends continuously on the vector s=(s_1,...,s_h) and on
+the multiplicities p_j, so the finite-block approximation is well
behaved.
-Spectral mesh limit. Approximate the limiting singular measure $nu$ by
+Spectral mesh limit. Approximate the limiting singular measure nu by
finitely many representative values, with mesh decreasing to zero and
-multiplicities equal to the mass assigned by $nu$. The no-outlier
+multiplicities equal to the mass assigned by nu. The no-outlier
assumption guarantees that the rectangular singular slots lie in the
-support of $nu$, so the representatives can be chosen positive
-throughout. Operator-norm continuity of $F_N$, together with the
-$o(M)$ dimension slack from structural zeros, transfers the
+support of nu, so the representatives can be chosen positive
+throughout. Operator-norm continuity of F_N, together with the
+o(M) dimension slack from structural zeros, transfers the
finite-block limits to the full problem. The polynomial rate of the
mesh approximation matches the operator-norm continuity constant, so
the two limits separate. This completes the proof of the main
theorem.
The mean convergence follows from boundedness: the pressure is
-uniformly bounded by $|f|_∞$, so probability convergence of
-$sup$ and $inf$ plus boundedness gives convergence of expectations.
+uniformly bounded by |f|_∞, so probability convergence of
+sup and inf plus boundedness gives convergence of expectations.
No separate argument is needed.
Extensions and boundaries. The main theorem constrains the aspect
@@ -1179,168 +1179,168 @@ ratio, the singular spectrum, and the activation. Several of these
constraints can be relaxed and one cannot.
Arbitrary aspect ratio. The limit formula holds for any deterministic
-$(M_N,N_N)$ with $M_N/N_Ntoalphain(0,∞)$. The angular and
+(M_N,N_N) with M_N/N_Ntoalphain(0,∞). The angular and
Dirichlet arguments use only limiting proportions. The additional case
-is $alpha=1$ with $N>M$ and $k=N-M=o(N)$ structural right-null
-coordinates. Let $B$ be the restriction of $A$ to the $M$ visible
-input coordinates, and let $Z_M(r)$ be the angular partition function
-of $B$ at radius $r$. If $b=|f|_∞$, the sphere in $mathbb R^N$bbR^N$
+is alpha=1 with N>M and k=N-M=o(N) structural right-null
+coordinates. Let B be the restriction of A to the M visible
+input coordinates, and let Z_M(r) be the angular partition function
+of B at radius r. If b=|f|_∞, the sphere in mathbb R^NbbR^N
decomposes as
-$$x=√(v),x'+u,qquad x'inmathbb R^M,bbR^M,quad uinmathbb R^k,$$bbR^k,$$
+x=√(v),x'+u, x'inmathbb R^M,bbR^M, uinmathbb R^k,bbR^k,
-with $v=|x|^2/NsimBeta(M/2,k/2)$ and $x'$ uniform on
-$S^{M-1}$ conditionally. The measured field at radius $√(Nv)$ has
-squared norm approximately $alpha s_N^2v$ when $s_N$ is the typical
-singular value. The endpoints $v$ small and $v$ near one contribute
+with v=|x|^2/NsimBeta(M/2,k/2) and x' uniform on
+S^{M-1} conditionally. The measured field at radius √(Nv) has
+squared norm approximately alpha s_N^2v when s_N is the typical
+singular value. The endpoints v small and v near one contribute
negligibly because the Beta density makes such values exponentially rare at
-the field-energy scale. Integration by parts over $v$ converts the
-sphere integral into a ball integral of $Z_M(r)$ over
-$r^2in[alpha s_-^2v_{rm lo},alpha s_+^2v_{rm hi}]$ times a
+the field-energy scale. Integration by parts over v converts the
+sphere integral into a ball integral of Z_M(r) over
+r^2in[alpha s_-^2v_{rm lo},alpha s_+^2v_{rm hi}] times a
Jacobian. The ball integral differs from the sphere integral by
-$frac12log$ of an explicit Beta-type factor, which is exactly the
-Dirichlet entropy of a null block of mass $1-alpha$. The same radial
-formula holds with $M$ denominators and the extra null block providing
+frac12log of an explicit Beta-type factor, which is exactly the
+Dirichlet entropy of a null block of mass 1-alpha. The same radial
+formula holds with M denominators and the extra null block providing
the endpoint.
Edge intervals and sublinear outliers. The assumption that every
-singular value stays in $supp,nu$ weakens as follows. Let
-$s_-=minsupp,nu$ and $s_+=maxsupp,nu$. The
+singular value stays in supp,nu weakens as follows. Let
+s_-=minsupp,nu and s_+=maxsupp,nu. The
requirement becomes
-$$max_{rlemin(M,N)}dist(s_{N,r},[s_-,s_+])to 0quadhbox{in probability}.$$
+max_{rlemin(M,N)}dist(s_{N,r},[s_-,s_+])to 0quadhbox{in probability}.
The edge interval suffices. If a sublinear number of singular values
-leak outside $[s_-,s_+]$, one or two extra Haar directions act as
+leak outside [s_-,s_+], one or two extra Haar directions act as
fake blocks. The leaked weight is assigned to those directions, and the negative
-Dirichlet entropy $frac12v_ilog(rho_i/v_i)$ with $v_i=o(1)$
-absorbs the effect as $rho_ito 0$. When $alpha<1$, the interval may
-be $[0,s_+]$, because the macroscopic input null space provides a
+Dirichlet entropy frac12v_ilog(rho_i/v_i) with v_i=o(1)
+absorbs the effect as rho_ito 0. When alpha<1, the interval may
+be [0,s_+], because the macroscopic input null space provides a
legitimate zero endpoint. Forced output zeros do not provide such an
-endpoint. The degenerate case $s_-=s_+=0$ gives
-$F_{alpha,nu}(f)=alpha f(0)$.
+endpoint. The degenerate case s_-=s_+=0 gives
+F_{alpha,nu}(f)=alpha f(0).
Unbounded activations with subquadratic growth. The theorem assumes
-$f$ bounded. It relaxes to $|f(z)|/(1+z^2)to 0$ with $f$ continuous.
+f bounded. It relaxes to |f(z)|/(1+z^2)to 0 with f continuous.
Conditional on input coefficients and singular values, Haar left
rotation makes the field a uniformly random vector on the sphere of
-radius $lambda_xsqrt M$ where $lambda_x=|Ax|/sqrt M$. On the
-good-norm event, $|A|_{op}le C$ and at least $kappa N$
-input directions have singular value at least $s_*>0$. For $L>0$
-there is small $lambda_0>0$ with
+radius lambda_xsqrt M where lambda_x=|Ax|/sqrt M. On the
+good-norm event, |A|_{op}le C and at least kappa N
+input directions have singular value at least s_*>0. For L>0
+there is small lambda_0>0 with
-$$sigma_N{lambda_x<lambda_0}le e^{-LM},$$
+sigma_N{lambda_x<lambda_0}le e^{-LM},
independently of the spectrum. Conditional on input coefficients and
-singular values, the density of any $k<M-2$ entries of the field on
-the sphere of radius $lambda_xsqrt M$ is
+singular values, the density of any k<M-2 entries of the field on
+the sphere of radius lambda_xsqrt M is
-$$frac{Gamma(M/2)}{Gamma((M-k)/2)(pi Mlambda_x^2)^{k/2}}(1-(|z|^2)/(Mlambda_x^2))_+^{(M-k-2)/2}.$$
+frac{Gamma(M/2)}{Gamma((M-k)/2)(pi Mlambda_x^2)^{k/2}}(1-(|z|^2)/(Mlambda_x^2))_+^{(M-k-2)/2}.
-For $kle M/2$ and $lambda_xgelambda_0$, this is at most
-$(C_0/lambda_0)^k$. If a measurable set $B$ has Lebesgue measure at
-most $delta$, then with $k=lceilgamma Mrceil$ and $0<gamma<1/2$,
+For kle M/2 and lambda_xgelambda_0, this is at most
+(C_0/lambda_0)^k. If a measurable set B has Lebesgue measure at
+most delta, then with k=lceilgamma Mrceil and 0<gamma<1/2,
-$$mathbb PbbP{#{r:(Ax)_rin B}ge k}lebinom{M}{k}(C_0delta/lambda_0)^k.$$
+mathbb PbbP{#{r:(Ax)_rin B}ge k}lebinom{M}{k}(C_0delta/lambda_0)^k.
-Choosing $R$ with $Λ^2/R^2legamma$ and approximating $f$ in
-Lebesgue measure on $[-R,R]$ by continuous $g$ with $|g|le b$ gives
+Choosing R with Λ^2/R^2legamma and approximating f in
+Lebesgue measure on [-R,R] by continuous g with |g|le b gives
the extension. The exceptional inputs have exponentially small
probability and negligible partition mass; on the remaining inputs the
-normalised energy difference is bounded by $ε+4bgamma$, which
+normalised energy difference is bounded by ε+4bgamma, which
tends to zero along a sequence of approximants.
Bounded Borel activations when the limiting spectrum is not a point
-mass at zero. A bounded Borel $f$ extends without continuity provided
-$nunedelta_0$. The proof uses the sphere-of-radius presentation and
-the small-field estimate. Set $b=|f|_∞$. Since
-$nunedelta_0$, the good-spectrum event has
-$|A|_{op}le C$ and at least $kappa N$ input directions
-with $sge s_*>0$. The small-field set has
-$sigma_N{lambda_x<lambda_0}le e^{-LM}$, and the field entries on
-the sphere satisfy the concentration bound with $C_0/lambda_0$. Any
-bounded Borel $f$ therefore approximates by bounded continuous $g$ up
+mass at zero. A bounded Borel f extends without continuity provided
+nunedelta_0. The proof uses the sphere-of-radius presentation and
+the small-field estimate. Set b=|f|_∞. Since
+nunedelta_0, the good-spectrum event has
+|A|_{op}le C and at least kappa N input directions
+with sge s_*>0. The small-field set has
+sigma_N{lambda_x<lambda_0}le e^{-LM}, and the field entries on
+the sphere satisfy the concentration bound with C_0/lambda_0. Any
+bounded Borel f therefore approximates by bounded continuous g up
to exponentially small partition mass.
Failure modes. The extensions are strict. Superquadratic activations
can make the pressure diverge even on a spectrum that converges to
zero, so the subquadratic growth hypothesis is needed. The bounded
-Borel extension fails when $nu=delta_0$: even the indicator
-${zne 0}$ can oscillate between two values along a sequence whose
+Borel extension fails when nu=delta_0: even the indicator
+{zne 0} can oscillate between two values along a sequence whose
rectangular singular slots converge weakly with no outlier in the
support of the limiting measure. Sublinear outliers that hide behind
structural output zeros break the edge interval hypothesis when the
zero endpoint is not a legitimate input null. Four worked counterexamples follow.
Counterexample: forced output zeros hide a small singular outlier.
-Let $M=1000N$ so $alpha=1000$. On even $N$ take $A_N$ equal to
-$sqrtalpha$ times a Haar orthonormal $Mtimes N$ frame; on odd $N$
+Let M=1000N so alpha=1000. On even N take A_N equal to
+sqrtalpha times a Haar orthonormal Mtimes N frame; on odd N
set one singular value to zero and keep the rest Haar. Let
-$f(z)=-cmin(z^2,1)$ with $cge 1$. On even $N$,
-$A=sqrtalpha U$ with $U$ a Haar orthonormal frame. For a unit input
-$u$, $sum_rf((Au)_r)=-csum_rmin(M(Uu)_r^2,1)$, and a Gaussian tail
-plus a small net on $S^{N-1}$ gives
-$inf_u(1/M)sum_rmin(M(Uu)_r^2,1)ge 1/80$ with probability at least
-$1-e^{-c_0M}$. So the even-$N$ pressure is at most $-calpha/80$,
-which is $-12.5c$ for $alpha=1000$. On odd $N$ one input direction
+f(z)=-cmin(z^2,1) with cge 1. On even N,
+A=sqrtalpha U with U a Haar orthonormal frame. For a unit input
+u, sum_rf((Au)_r)=-csum_rmin(M(Uu)_r^2,1), and a Gaussian tail
+plus a small net on S^{N-1} gives
+inf_u(1/M)sum_rmin(M(Uu)_r^2,1)ge 1/80 with probability at least
+1-e^{-c_0M}. So the even-N pressure is at most -calpha/80,
+which is -12.5c for alpha=1000. On odd N one input direction
maps to zero; a uniform input decomposes as
-$sqrt Vu_{rm live}+√(1-V)u_{rm null}$ with
-$VsimBeta((N-1)/2,1/2)$. The event $Vle v$ has
-log-probability per $N$ tending to $frac12log v$, and on this event
-the energy is at least $-cv$, giving
+sqrt Vu_{rm live}+√(1-V)u_{rm null} with
+VsimBeta((N-1)/2,1/2). The event Vle v has
+log-probability per N tending to frac12log v, and on this event
+the energy is at least -cv, giving
-$${rm pressure}ge tfrac12log v -cv = -0.5log 160-6.25capprox -2.5-6.25c.$$
+{rm pressure}ge tfrac12log v -cv = -0.5log 160-6.25capprox -2.5-6.25c.
-For $c=1$ the even-$N$ pressure is at most $-12.5$ while the odd-$N$
-pressure is at least $-8.75$, so the limits differ. The rectangular
+For c=1 the even-N pressure is at most -12.5 while the odd-N
+pressure is at least -8.75, so the limits differ. The rectangular
singular measure excluding the structural zero still converges to
-$delta_{sqrtalpha}$, but the output eigenvalue measure hides the
+delta_{sqrtalpha}, but the output eigenvalue measure hides the
outlier in the zero mass. No-outlier control must be on the rectangular
singular slots, not on output eigenvalues. The uniform lower bound
-$inf_u Q(u)ge 1/80$ is a Kashin-type spreading estimate for Haar
+inf_u Q(u)ge 1/80 is a Kashin-type spreading estimate for Haar
frames; a norm-comparison restatement in the lasso analysis of Bayati
and Montanari gives the same conclusion with explicit constants.
./pdf/bayati-montanari-lasso.pdf
-Counterexample: unbounded limiting spectrum. Let $d_k=exp(8^k)$ and
-$nu=sum_{kge 1}2^{-k}delta_{d_k}$ with unbounded support. For
-large $N$ choose $K=lfloorlog_2log Nrfloor$ and multiplicities
-$n_j=lfloor N2^{-j}rfloor$, with the top block about $N/2$. Let $f$
-be continuous on $[0,1]$ equal to $c_Kin{0,1}$ on
-$[d_K^{1/2},d_K^2]$, alternating in $c_K$ with $K$, interpolating in
-the gaps. The top block has singular value $d_K=exp(8^K)$ and its
-field values are of size $d_K$, so on the top block $f$ evaluates to
-$c_K$ almost surely; smaller blocks contribute negligibly. The
-pressure along the sub-$K$ sequence alternates between the values
-$c_Kin{0,1}$. Weak convergence to $nu$ plus the outlier bound is
-not enough when $nu$ has unbounded support; compact support is
+Counterexample: unbounded limiting spectrum. Let d_k=exp(8^k) and
+nu=sum_{kge 1}2^{-k}delta_{d_k} with unbounded support. For
+large N choose K=lfloorlog_2log Nrfloor and multiplicities
+n_j=lfloor N2^{-j}rfloor, with the top block about N/2. Let f
+be continuous on [0,1] equal to c_Kin{0,1} on
+[d_K^{1/2},d_K^2], alternating in c_K with K, interpolating in
+the gaps. The top block has singular value d_K=exp(8^K) and its
+field values are of size d_K, so on the top block f evaluates to
+c_K almost surely; smaller blocks contribute negligibly. The
+pressure along the sub-K sequence alternates between the values
+c_Kin{0,1}. Weak convergence to nu plus the outlier bound is
+not enough when nu has unbounded support; compact support is
essential.
-Counterexample: bounded Borel at zero spectrum. Let $M=N$ and set
-$A_N=0$ on even $N$ and $A_N=s_NO_N$ on odd $N$ with $s_Nto 0$ and
-$O_N$ Haar orthogonal. Let $f(z)=1_{{zne 0}}$. On even $N$ the
-field is zero, $f(0)=0$, so the pressure is zero. On odd $N$ the field
+Counterexample: bounded Borel at zero spectrum. Let M=N and set
+A_N=0 on even N and A_N=s_NO_N on odd N with s_Nto 0 and
+O_N Haar orthogonal. Let f(z)=1_{{zne 0}}. On even N the
+field is zero, f(0)=0, so the pressure is zero. On odd N the field
is a nonzero multiple of a Haar vector, so all coordinates are nonzero
-almost surely, $f((A_Nx)_r)=1$ for every $r$, and the pressure is
-$1$. Both spectral sequences have rectangular singular slots
-converging to $delta_0$ with no outliers. Bounded Borel activations
-with $nu=delta_0$ fail.
+almost surely, f((A_Nx)_r)=1 for every r, and the pressure is
+1. Both spectral sequences have rectangular singular slots
+converging to delta_0 with no outliers. Bounded Borel activations
+with nu=delta_0 fail.
Counterexample: continuous activation without a growth bound. Let
-$M=N$ and set $A_N=0$ on even $N$, $A_N=N^{-1/4}O_N$ on odd $N$ with
-$O_N$ Haar. Let $f(z)=z^8$. On even $N$ the pressure is zero. On odd
-$N$, $|A_N|_{op}=N^{-1/4}$ and
-$sum_r(A_Nx)_r^8=N^{-2}sum_rΩ_r^8$ with $Ω=O_Nx$. The
-coordinate event $Ω_1^2in[N/4,N/2]$ has probability at least
-$frac18 2^{-(N-3)/2}$ and on it $Ω_1^8ge N^4/256$, so
+M=N and set A_N=0 on even N, A_N=N^{-1/4}O_N on odd N with
+O_N Haar. Let f(z)=z^8. On even N the pressure is zero. On odd
+N, |A_N|_{op}=N^{-1/4} and
+sum_r(A_Nx)_r^8=N^{-2}sum_rΩ_r^8 with Ω=O_Nx. The
+coordinate event Ω_1^2in[N/4,N/2] has probability at least
+frac18 2^{-(N-3)/2} and on it Ω_1^8ge N^4/256, so
-$$sum_r(A_Nx)_r^8ge N^2/256,$$
+sum_r(A_Nx)_r^8ge N^2/256,
and therefore
-$${rm pressure}ge frac1N[(N^2)/(256)+log(frac18 2^{-(N-3)/2})]=frac N{256}-frac12log 2-(3log 2)/(2N)to+∞.$$
+{rm pressure}ge frac1N[(N^2)/(256)+log(frac18 2^{-(N-3)/2})]=frac N{256}-frac12log 2-(3log 2)/(2N)to+∞.
-The pressure diverges on odd $N$ while it is zero on even $N$, and both
-spectra converge to $delta_0$ with no outliers. Continuous activation
+The pressure diverges on odd N while it is zero on even N, and both
+spectra converge to delta_0 with no outliers. Continuous activation
without a growth bound diverges.