diff b34550f..b1f9bc4 spherical_perceptron_bi_orthogonal_v2.txt
diff --git a/spherical_perceptron_bi_orthogonal_v2.txt b/spherical_perceptron_bi_orthogonal_v2.txt
index 8152060..dc7565d 100644
--- a/spherical_perceptron_bi_orthogonal_v2.txt
+++ b/spherical_perceptron_bi_orthogonal_v2.txt
@@ -1,13 +1,13 @@
-Let $A$ be a real $M\times N$ matrix, and let $\sigma_N$ denote uniform
-probability measure on the sphere $S^N=\{x\in\mathbb R^N:\|x\|^2=N\}$.
-For a bounded continuous $f:\mathbb R\to\mathbb R$, 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
-$$F_N(A,f)=\frac1N\log\int_{S^N}\exp{\sum_{r=1}^M f((Ax)_r)}d\sigma_N(x).$$
+$$F_N(A,f)=frac1Nlogint_{S^N}exp{sum_{r=1}^M f((Ax)_r)}dsigma_N(x).$$
The measurement matrix $A$ carries a disorder distribution on real
-$M\times N$ matrices. A temperature parameter enters by writing
-$f=\beta\phi$. The limit of $F_N(A,f)$ as $M/N\to\alpha>0$ for a
+$Mtimes N$ 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
@@ -134,24 +134,24 @@ 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 $[0,∞)$.
+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 $\alpha>0$, $M=\lfloor\alpha N\rfloor$, and $f\in C_b(\mathbb R)$.
+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 $A_N\overset{d}{=}UA_NV$ for any independent
-uniform orthogonal $U\in O(M)$, $V\in O(N)$, and suppose the empirical
-singular-value measure $r_N^{-1}\sum_{r\le r_N}\delta_{s_{N,r}}$
+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
-$\max_{r\le r_N}dist(s_{N,r},supp\,\nu)\to 0$ in
+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 $F_{\alpha,\nu}(f)$.
+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
@@ -159,105 +159,105 @@ 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
-$n_j/M\to p_j>0$. 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 $\mathcal I=\{1,...,h\}$ when $\alpha\ge 1$ and
-$\mathcal I=\{0,1,...,h\}$ when $\alpha<1$. For each block
-$i\in\mathcal I$ let $\rho_i$ denote the fraction of the input squared
-norm assigned to block $i$, with $\rho_i>0$ and $\sum_i\rho_i=1$. The
+$n_j/Mto p_j>0$. 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
value for this finite approximation is
-$$\sup_{\rho}{\frac12\sum_{i\in\mathcal I}v_i\log(\rho_i)/(v_i)+\alpha P_p^f(((s_j^2\rho_j)/(\alpha))_{j=1}^h)},$$
+$$sup_{rho}{frac12sum_{iinmathcal I}calIv_ilog(rho_i)/(v_i)+alpha P_p^f(((s_j^2rho_j)/(alpha))_{j=1}^h)},$$
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 $F_{\alpha,\nu}(f)$.
+approximations as the spectral mesh tends to zero is $F_{alpha,nu}(f)$.
-The term $\frac12\sum v_i\log(\rho_i/v_i)$ 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-$\mathbb R^M$ angular
-pressure into a per-$\mathbb R^N$ pressure.
+$v_iN$. The factor $alpha$ converts the per-$mathbb R^M$bbR^M$ angular
+pressure into a per-$mathbb R^N$bbR^N$ pressure.
-Let $E_1,...,E_h\subset\mathbb R^M$ be mutually orthogonal Haar
-subspaces with $\dim E_j=n_j$ and $n_j/M\to p_j>0$, so $p=\sum_jp_j\le 1$.
-For $a_j\ge 0$, let $\sigma_{E_j,Ma_j}$ denote uniform probability
-measure on the sphere $\{y\in E_j:\|y\|^2=Ma_j\}$ of squared radius
+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
angular pressure
-$$P_M^f(a)=\frac1M\log\int\exp{\sum_{r=1}^M f(\sum_jy_{j,r})}\prod_j d\sigma_{E_j,Ma_j}(y_j).$$
+$$P_M^f(a)=frac1Mlogintexp{sum_{r=1}^M f(sum_jy_{j,r})}prod_j dsigma_{E_j,Ma_j}(y_j).$$
The subspaces are held fixed inside the integral, and the vectors $y_j$
are integrated independently. The angular theorem asserts that
$P_M^f(a)$ converges in probability, uniformly on compact sets of
-$a\in[0,∞)^h$, to a deterministic continuous function $P_p^f(a)$.
+$ain[0,∞)^h$, to a deterministic continuous function $P_p^f(a)$.
For $p<1$ and positive $a$, the limit equals
-$$P_p^f(a)=J_p-H_p(a)+\lim_{\lambda\to∞}\lim_{B\to∞}\lim_{\theta↓ 0}\mathcal B_{p,a}(\kappa,\lambda,B,\theta),$$
+$$P_p^f(a)=J_p-H_p(a)+lim_{lambdato∞}lim_{Bto∞}lim_{theta↓ 0}mathcal B_calB_{p,a}(kappa,lambda,B,theta),$$
-for every sufficiently large fixed $\kappa$, with
+for every sufficiently large fixed $kappa$, with
-$$s_a=\sum_ja_j,\qquad H_p(a)=\frac12\sum_jp_j\log(2\pi e a_j)/(p_j),\qquad J_p=\frac12\int_0^p\log(1-u)\,du.$$
+$$s_a=sum_ja_j,qquad H_p(a)=frac12sum_jp_jlog(2pi e a_j)/(p_j),qquad J_p=frac12int_0^plog(1-u),du.$$
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.
+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
-$d_j\times d_j$ matrix with positive diagonal and every column norm at
-most $\kappa$. For a row vector $\xi_j\in\mathbb R^{d_j}$, label the
-entries of $\xi_j^TR_j$ by $x,j$ and $z_i,j$ for $j\le i<h$. Write
+$d_jtimes d_j$ 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
-$$y=\sum_j(\xi_j^TR_j)_{x,j},\qquad u_i=\sum_{j\le i}(\xi_j^TR_j)_{z_i,j}.$$
+$$y=sum_j(xi_j^TR_j)_{x,j},qquad u_i=sum_{jle i}(xi_j^TR_j)_{z_i,j}.$$
The labels $y,u_i$ refer to single coordinates of the row fields in the
Gaussian model. Set
-$$g_1=y^2,\ c_1=s_a;\qquad g_2=-\min(y^2,B),\ c_2=-s_a;\qquad g_{i+2}=(y-u_i)^2,\ c_{i+2}=\sum_{j>i}a_j.$$
+$$g_1=y^2, c_1=s_a;qquad g_2=-min(y^2,B), c_2=-s_a;qquad g_{i+2}=(y-u_i)^2, c_{i+2}=sum_{j>i}a_j.$$
Define
-$$\mathcal B_{p,a}(\kappa,\lambda,B,\theta)=\sup_{R_1,...,R_h}[\sum_jp_j(\log(\kappa√((2\pi e)/(p_j)))+\log\frac{R_{j,11}}\kappa)+\theta\sum_{\ell=2}^{d_j}p_j\log\frac{R_{j,\ell\ell}}\kappa+\theta\min_{0\le t_\ell\le\lambda/\theta}{t\cdot c+\mathcal G_{\theta,p}((f(y))/(\theta)-t\cdot g)}],$$
+$$mathcal B_calB_{p,a}(kappa,lambda,B,theta)=sup_{R_1,...,R_h}[sum_jp_j(log(kappa√((2pi e)/(p_j)))+logfrac{R_{j,11}}kappa)+thetasum_{ell=2}^{d_j}p_jlogfrac{R_{j,ellell}}kappa+thetamin_{0le t_elllelambda/theta}{tcdot c+mathcal G_calG_{theta,p}((f(y))/(theta)-tcdot g)}],$$
-where $\mathcal G_{\theta,p}$ 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 $p_j\theta$ 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 $\lambda\to∞$. The truncation $B$ is removed last.
+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 $d_j\ge 1$ and a positive exponent
-$0<\theta<1$. Let $n_j/M\to p_j>0$ and let
+arbitrary column counts $d_jge 1$ and a positive exponent
+$0<theta<1$. Let $n_j/Mto p_j>0$ and let
-$$X_j\inSt(n_j,d_j):=\{X\in\mathbb R^{n_j\times d_j}:X^TX=n_jI_{d_j}\}$$
+$$X_jinSt(n_j,d_j):={Xinmathbb R^bbR^{n_jtimes d_j}:X^TX=n_jI_{d_j}}$$
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 $g_{r,j}\in\mathbb R^{n_j}$, define the row
+standard Gaussian vectors $g_{r,j}inmathbb R^bbR^{n_j}$, define the row
fields
-$$\xi_{r,j}(X)=\frac{X_j^Tg_{r,j}}{√(n_j)},\qquad 1\le r\le M.$$
+$$xi_{r,j}(X)=frac{X_j^Tg_{r,j}}{√(n_j)},qquad 1le rle M.$$
For a Hamiltonian $H$ define
-$$\mathcal Z_{M,\theta}(H)=[\int(\int e^H\,d\mu_{in})^\theta d\mu_{out}]^{1/\theta}.$$
+$$mathcal Z_calZ_{M,theta}(H)=[int(int e^H,dmu_{in})^theta dmu_{out}]^{1/theta}.$$
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
@@ -265,44 +265,44 @@ normalized log integral.
Choose a common finite grid
-$$0=\zeta_0<\zeta_1<...<\zeta_k=1,\qquad w_i=\zeta_{i+1}-\zeta_i,\quad 0\le i<k,$$
+$$0=zeta_0<zeta_1<...<zeta_k=1,qquad w_i=zeta_{i+1}-zeta_i,quad 0le i<k,$$
-containing $\theta$. A trial in block $j$ consists of symmetric matrices
+containing $theta$. A trial in block $j$ consists of symmetric matrices
-$$0\preceq q_{j,0}\preceq...\preceq q_{j,k-1}\preceq I_{d_j},\qquad q_{j,-1}=0,\quad q_{j,k}=I_{d_j},$$
+$$0preceq q_{j,0}preceq...preceq q_{j,k-1}preceq I_{d_j},qquad q_{j,-1}=0,quad q_{j,k}=I_{d_j},$$
-with the pinning condition $q_{j,i}e_1=e_1$ whenever $\zeta_i\ge\theta$.
-Write $q_j(u)=q_{j,i}$ on $[\zeta_i,\zeta_{i+1})$. Let
-$F_{j,i}=\mathbb R^{d_j}$ 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
-$$\mathsf K_{j,i}=\sum_{r=i+1}^k\zeta_r(q_{j,r}-q_{j,r-1}^{\phantom{}}),\qquad \mathsf P_{j,i}=(\mathsf K_{j,i}|_{F_{j,i}})^{-1}.$$
+$$mathsf K_sfK_{j,i}=sum_{r=i+1}^kzeta_r(q_{j,r}-q_{j,r-1}^{phantom{}}),qquad mathsf P_sfP_{j,i}=(mathsf K_sfK_{j,i}|_{F_{j,i}})^{-1}.$$
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
-$\mathsf K_{j,i}^0$ analogously. The block entropy is
+$mathsf K_sfK_{j,i}^0$ analogously. The block entropy is
-$$2S_\theta(q_j)=tr(\mathsf P_{j,0}q_{j,0})+\sum_{i=1}^k\frac1{\zeta_i}\log(\frac{\det\mathsf K_{j,i-1}}{\det\mathsf K_{j,i}}\frac{\det\mathsf K_{j,i}^0}{\det\mathsf K_{j,i-1}^0}),$$
+$$2S_theta(q_j)=tr(mathsf P_sfP_{j,0}q_{j,0})+sum_{i=1}^kfrac1{zeta_i}log(frac{detmathsf K_sfK_{j,i-1}}{detmathsf K_sfK_{j,i}}frac{detmathsf K_sfK_{j,i}^0}{detmathsf K_sfK_{j,i-1}^0}),$$
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 $D:\prod_j\mathbb R^{d_j}\to\mathbb R$
-satisfying $-C(1+|z|^2)\le D(z)\le C$ for some $C<∞$, let
+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
block-diagonal covariances
$diag_j(q_{j,i}-q_{j,i-1})$. Set $U_k=D$ and define backwards,
for $i=k,k-1,...,1$,
-$$U_{i-1}(z)=\frac1{\zeta_i}\log\mathbb E\exp\{\zeta_iU_i(z+Z_i)\}.$$
+$$U_{i-1}(z)=frac1{zeta_i}logmathbb EbbEexp{zeta_iU_i(z+Z_i)}.$$
-The single-pattern value is $V_D(q)=\mathbb EU_0(Z_0)$; 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
-$$\mathcal G_{\theta,p}(D)=\inf_q{\sum_{j=1}^hp_jS_\theta(q_j)+V_D(q)},$$
+$$mathcal G_calG_{theta,p}(D)=inf_q{sum_{j=1}^hp_jS_theta(q_j)+V_D(q)},$$
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
@@ -310,24 +310,24 @@ 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
-$0<\theta<1$, $n_j/M\to p_j>0$, 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
-$$\frac1M\log\mathcal Z_{M,\theta}(\sum_{r=1}^MD(\xi_r))→\mathcal G_{\theta,p}(D)$$
+$$frac1Mlogmathcal Z_calZ_{M,theta}(sum_{r=1}^MD(xi_r))→mathcal G_calG_{theta,p}(D)$$
-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
-$\bar g=M^{-1}\sum_rg(\xi_r)$, and fix $d>0$ and $c\in\mathbb R^s$.
+$barbar g=M^g=M^{-1}sum_rg(xi_r)$, and fix $d>0$ and $cinmathbb R^s$.bbR^s$.
The pressure for the soft-penalised Hamiltonian
-$$H=\sum_rD(\xi_r)-Md\sum_{\ell=1}^s(\bar g_\ell-c_\ell)_+$$
+$$H=sum_rD(xi_r)-Mdsum_{ell=1}^s(barbar g_g_ell-c_ell)_+$$
converges in probability to
-$$\min_{t\in[0,d]^s}\{t\cdot c+\mathcal G_{\theta,p}(D-t\cdot g)\}.$$
+$$min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.$$
-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
@@ -335,34 +335,34 @@ 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 $\mathcal B_{p,a}$ the
-row tests $g_\ell$ are nonnegative quadratics, and the contributions
-$-t\cdot g$ 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
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 $(1/\theta)\log\int(\int e^Hd\mu_{\rm in})^\theta d\mu_{\rm out}$.
+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
+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 $q$.
-Because $\mathcal Z_{M,\theta}(H)$ 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
-$M^{-1}\log\mathcal Z_{M,\theta}(H)$ 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.
@@ -370,11 +370,11 @@ Overlaps and matrix synchronization. Write replicas as superscripts
$a,b$. The scalar and matrix overlaps of a configuration $(X,A)$ at
the disorder level are
-$$\mathcal R_j^{ab}=((X_j^a)^TX_j^b)/(n_j).$$
+$$mathcal R_j^calR_j^{ab}=((X_j^a)^TX_j^b)/(n_j).$$
The array indexed jointly by replica and column is Gram and the self
blocks are fixed at $I$. The perturbing scalarizations are
-$v^T(\mathcal R_j^{ab})^{\circ\ell}v$ for $\ell=1,2$ and a countable
+$v^T(mathcal R_j^calR_j^{ab})^{circell}v$ 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
@@ -386,11 +386,11 @@ 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,
+$psi$ of the generated scalarization-and-label vector on one edge,
-$$\mathbb E< F\psi(T^{1,r+1})>=\frac1r\mathbb E< F>\,\mathbb E<\psi(T^{12})>+\frac1r\sum_{a=2}^r\mathbb E< F\psi(T^{1a})>,$$
+$$mathbb E<bbE< Fpsi(T^{1,r+1})>=frac1rmathbb E<bbE< F>,mathbb E<bbE<psi(T^{12})>+frac1rsum_{a=2}^rmathbb E<bbE< Fpsi(T^{1a})>,$$
-where brackets denote Gibbs expectation and $\mathbb E$ averages all
+where brackets denote Gibbs expectation and $mathbb E$bbE$ 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
@@ -402,9 +402,9 @@ 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 $\mathcal R_j^{12}$ were not symmetric in the limit, dense
-first-power forms $\alpha$ and antisymmetric perturbations $\beta$ of
-$\mathcal R_j^{12}$ would generate crossed values on edges $12$ and
+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.
@@ -422,16 +422,16 @@ 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
-$0\preceq q_j\preceq I$. The label coordinate retains its prescribed
-atom masses, so its sharing threshold sits at quantile $\theta$. When
-the first column is shared, $\mathcal R_j^{ab}e_1=e_1$; hence the
+$0preceq q_jpreceq I$. 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
step variation,
-$$\left.(d)/(dt)V_D(q+t\dot q)|_{t=0}=-\frac12\sum_{j,i}w_i\,tr(\dot q_{j,i}\mathbb E[M_{j,i}M_{j,i}^T]),$$
+$$left.(d)/(dt)V_D(q+tdot q)|_{t=0}=-frac12sum_{j,i}w_i,tr(dot q_{j,i}mathbb E[M_bbE[M_{j,i}M_{j,i}^T]),$$
where $M_{j,i}$ is the conditional mean of the terminal $j$-gradient
under the successive tilted kernels. Gaussian covariance
@@ -439,109 +439,109 @@ 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 $G_{j,i}=\mathbb E[M_{j,i}M_{j,i}^T]$ are nondecreasing
-and uniformly bounded. Put $T_{j,i}=\sum_{r=i}^{k-1}w_r(\widetilde
+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
-$$\int_t^1\widetilde q_j(u)du\succeq\int_t^1q_j(u)du\quad\hbox{for every }j,t\ ⇒\ V_D(\widetilde q)\le V_D(q).$$
+$$int_t^1widetilde q_j(u)dusucceqint_t^1q_j(u)duquadhbox{for every }j,t ⇒ V_D(widetilde q)le V_D(q).$$
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 $0\preceq b_0\preceq... \preceq b_{k-1}$
+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
-$$limsup_{n\to∞}\ell_n(b)\le S_\theta(q)-\frac12\sum_iw_itr(q_ib_i)$$
+$$limsup_{nto∞}ell_n(b)le S_theta(q)-frac12sum_iw_itr(q_ib_i)$$
for every eligible $q$, locally uniformly in the field levels.
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 $d\times d$ multipliers. Define
+over symmetric $dtimes d$ multipliers. Define
-$$\Psi(C,b)=\frac12tr(Cb_0^{\phantom{.}})+\hbox{(expected tilted energy functionals)}$$
+$$Psi(C,b)=frac12tr(Cb_0^{phantom{.}})+hbox{(expected tilted energy functionals)}$$
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
+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 $\tfrac12I\preceq Q\preceq\tfrac32 I$,
+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 $Q_1,Q_{F\mid1}$ for
+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
-$$\theta^{-1}Ent(Q_1\mid\gamma_1)+\mathbb E_{Q_1}Ent(Q_{F\mid1}\mid\gamma_{d-1})\le\theta^{-1}Ent(N(0,Q)\mid\gamma_d),$$
+$$theta^{-1}Ent(Q_1midgamma_1)+mathbb E_bbE_{Q_1}Ent(Q_{Fmid1}midgamma_{d-1})letheta^{-1}Ent(N(0,Q)midgamma_d),$$
and this permits correlated trial coordinates. Hence
-$$\Psi(C,b)-\tfrac12tr\,C\ge\tfrac12tr(C(Q-I))-\frac1{2\theta}(tr\,Q-\log\det Q-d).$$
+$$Psi(C,b)-tfrac12tr,Cgetfrac12tr(C(Q-I))-frac1{2theta}(tr,Q-logdet Q-d).$$
-Choosing $Q=I+\tfrac12sign(C)$ yields
+Choosing $Q=I+tfrac12sign(C)$ yields
-$$\Psi(C,b)-\tfrac12tr\,C\ge\tfrac14\|C\|_*-c_d/\theta,$$
+$$Psi(C,b)-tfrac12tr,Cgetfrac14|C|_*-c_d/theta,$$
-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
+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
+$Psi(C,b)-tr,C/2$ 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 $\mathbb Ess^T=I$.
+gives the tilted self-covariance identity $mathbb Ess^T=I$.bbEss^T=I$.
-Under the tilted kernels, let $\mathcal H_i$ be the sigma-field
-generated by the field increments up to $\zeta_i$ plus the shared spin
-coordinates when $\zeta_i\ge\theta$, and let $M_i=\mathbb E[s\mid
-\mathcal H_i]$. Let $\mathsf K_i^*$ denote the Hessian of $U_i$ in its
+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
single log-exponential integration gives the recursion
-$$\mathsf K_{i-1}^*=\mathsf K_i^*+\zeta_iCov(M_i\mid\mathcal H_{i-1}),$$
+$$mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_iCov(M_imidmathcal H_calH_{i-1}),$$
with deterministic conditional covariances. Averaging and the
martingale property yield
-$$q_i^*=\mathbb E[M_iM_i^T],\qquad \mathsf K_k^*=0,\qquad \mathsf K_{i-1}^*=\mathsf K_i^*+\zeta_i(q_i^*-q_{i-1}^*),\qquad q_0^*=\mathsf K_0^*b_0\mathsf K_0^*.$$
+$$q_i^*=mathbb E[M_iM_i^T],bbE[M_iM_i^T],qquad mathsf K_k^*=0,sfK_k^*=0,qquad mathsf K_sfK_{i-1}^*=mathsf K_i^*+sfK_i^*+zeta_i(q_i^*-q_{i-1}^*),qquad q_0^*=mathsf K_0^*b_0sfK_0^*b_0mathsf K_0^*.$$sfK_0^*.$$
-The inverse identity $\mathsf K_i\mathsf P_i\mathsf K_i=\mathsf K_i$
-on the active space, combined with the pinning across $\theta$,
+The inverse identity $mathsf K_isfK_imathsf P_isfP_imathsf K_i=sfK_i=mathsf K_i$sfK_i$
+on the active space, combined with the pinning across $theta$,
identifies $q_0^*$ with the precision condition. The final statement
is the canonical duality: the minimiser $C(b)$ of
-$\Psi(C,b)-tr\,C/2$ yields
+$Psi(C,b)-tr,C/2$ yields
-$$J(b):=\Psi(C(b),b)-\tfrac12tr(C(b)+b_{k-1})=S_\theta(q^*)-\frac12\sum_iw_itr(q_i^*b_i)=\inf_q{S_\theta(q)-\frac12\sum_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)},$$
and $q^*$ is the unique minimiser of the right side. The path $q^*$ is
-eligible. For fixed $\theta,d$ and bounded $b_{k-1}$, all such
+eligible. For fixed $theta,d$ and bounded $b_{k-1}$, all such
multipliers are bounded uniformly over the grids. The tilted terminal
field has uniformly bounded second and fourth moments.
Uniqueness and value identification. The entropy functional
-$S_\theta$ is strictly convex in the path in the
-$2S=trPq+...$ sense: expanding $A_{i-1}=A_i+\zeta_i(e_{i-1}-e_i)$
+$S_theta$ is strictly convex in the path in the
+$2S=trPq+...$ sense: expanding $A_{i-1}=A_i+zeta_i(e_{i-1}-e_i)$
successively cancels cross terms and leaves a sum of squares. In
-addition $S_\theta(q^0)=0$ and the gradient at $q^0$ vanishes; hence
-$S_\theta\ge 0$. At the canonical path, $\mathsf P_0q_0^*\mathsf P_0=b_0$.
+addition $S_theta(q^0)=0$ and the gradient at $q^0$ vanishes; hence
+$S_thetage 0$. At the canonical path, $mathsf P_0q_0^*sfP_0q_0^*mathsf P_0=b_0$.sfP_0=b_0$.
Compress the inverse identity to each nested $F_i$ and telescope; the
scaled entropy gradient in the rightmost expression becomes
$b_i|_{F_i}$. Uniqueness of the minimiser then holds.
The dual values agree. Vary the field levels along the segment from
-zero to $b$. Covariance differentiation of $\Psi$ before subtracting
-the diagonal shift inserts $\tfrac12(\mathsf K_i^*+\zeta_i\mathbb
-EM_iM_i^T)$ for an increment at level $i$. Telescoping and
-$\mathbb Ess^T=I$ give
+zero to $b$. Covariance differentiation of $Psi$ before subtracting
+the diagonal shift inserts $tfrac12(mathsf K_i^*+sfK_i^*+zeta_imathbb
+EM_iM_i^T)$bbEM_iM_i^T)$ for an increment at level $i$. Telescoping and
+$mathbb Ess^T=I$bbEss^T=I$ give
-$$\dot J=-\tfrac12\sum_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
@@ -552,19 +552,19 @@ 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
uniformly small, and Gaussian interpolation bounds the log change by
-half the diagonal-minus-pair trace difference. Letting $n\to∞$
+half the diagonal-minus-pair trace difference. Letting $nto∞$
and then shrinking the neighborhood gives a lower bound on the
-canonical integral, so $limsup_n\ell_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$.
Upper bound for the Gaussian theorem. Fix a trial path $q$ in the
-Gaussian functional $\mathcal G_{\theta,p}$. The proof produces an
-expected pressure upper bound that matches $\mathcal G_{\theta,p}$ at
+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
@@ -575,27 +575,27 @@ nonlinear pattern Hamiltonian.
Enrich the model. Add independent linear Gaussian fields with
covariance levels tied to $q$. Replace the deterministic pattern
-count $M$ by Poisson$(Mr)$ with $r\in[0,1]$, so that $r=0$ gives no
+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=r\sqrt 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 E\mathcal P_{M,r}(b,u)+\sum_jp_jS_\theta(\tilde q_{j})+\frac r{2}\sum_iw_itr(\tilde 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_{M,r}$ is the expected (perturbed, enriched)
-pressure, $\tilde q_j$ is the covariance of the linear field, $b$ is
+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
+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_{\theta,p}$ at that trial by $3\delta$ for some
-$\delta>0$. Then $A_{M,0}(0,u)$ is $O(e_M^2)$ and $A_{M,1}(b_{r=1},u)$
+$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
parameters, and the field cap not binding. At this minimiser the
@@ -603,14 +603,14 @@ 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
+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 $\tilde b\succeq b$ of the linear-field path
-with compact support, the covariance comparison $V_D(\tilde q)\le
+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
+$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
@@ -618,10 +618,10 @@ $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 $M\to∞$, 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}\log\mathcal Z_{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
@@ -634,8 +634,8 @@ 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 $\tilde q$ with the trial
-$q$, so the contact-point upper bound matches $\mathcal G_{\theta,p}$
+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.
@@ -648,17 +648,17 @@ 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)=\eta\sum_rtr(\Delta b_r\sum_{i\ge r}w_i(I-q_i))\ge c_q\sum_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=\eta\min_{r:F_r\ne\{0\}}\{w_r\lambda_{\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_r\succeq\mathsf K_r\succ 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
functional is positive at a sufficiently large fixed trace cap. Thus a
@@ -680,22 +680,22 @@ 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
+$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
+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
+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
@@ -703,18 +703,18 @@ by $O(s_M^2/M)$, and integrating the second derivative controls
thermal variance. Integrated convex difference quotients with step
$s$ give
-$$C_a\,s\,s_M^2/M+C/(s\sqrt M)$$
+$$C_a,s,s_M^2/M+C/(ssqrt M)$$
for the integrated absolute fluctuation of the first derivative. At
$s=M^{-1/8}$ this implies
-$$\int\mathbb E<|Y_a-\mathbb E< Y_a>|>/s_M\,d\pi\to 0.$$
+$$intmathbb E<|Y_a-bbE<|Y_a-mathbb E<bbE< Y_a>|>/s_M,dpito 0.$$
The estimates hold on a slightly larger coupling interval so endpoint
difference quotients cause no difficulty, and a diagonal selection
over countably many coordinates gives the good sets.
-Restoring patterns. Fix a batch of $\ell$ fresh independent patterns.
+Restoring patterns. Fix a batch of $ell$ fresh independent patterns.
In the enlarged model, draw spin columns by Gram-Schmidt: the first
column is drawn at its prescribed vertex, and the remaining columns
are drawn at visits. Restrict all normalised triangular lengths and
@@ -724,7 +724,7 @@ probability tending to one independently of the preceding frame, and
the recursion normalisation costs are $o(n)$.
The enlarged Gibbs measure with the $M$ original patterns and the
-$\ell$ fresh ones has self covariance close to the identity and cross
+$ell$ fresh ones has self covariance close to the identity and cross
covariance close to the empirical overlap of the sampled pair. This is
the physical prediction from spatial rotation invariance; it holds in
the enlarged Gibbs law for each fixed coordinate, uniformly over the
@@ -739,35 +739,35 @@ Gaussian columns proves the conditional version.
Fix small $ε>0$ and restrict the enlarged partition function to
the thick shell
-$$\mathcal I=\{\|Z_j^TZ_j/r_j-I\|\leε\hbox{ for every }j\}.$$
+$$mathcal I=calI={|Z_j^TZ_j/r_j-I|leεhbox{ for every }j}.$$
This only lowers its logarithm. On the shell, replace the larger
perturbation by the base perturbation at the same $u$. For fixed
-$\ell$, overlaps differ by $O_\ell(M^{-1})$, and the summable kernel
+$ell$, overlaps differ by $O_ell(M^{-1})$, and the summable kernel
Lipschitz constants, the amplitude $s_M$, and the exponentially small
-coefficient tail together imply an $O_\ell(M^{-1/4})+o(1)$ covariance
+coefficient tail together imply an $O_ell(M^{-1/4})+o(1)$ covariance
error in unnormalised logarithms. Gaussian interpolation gives a
vanishing cost both for expected logs and for bounded replica tests.
On the shell, the physical cross covariance of two fresh patterns
matches the empirical matrix overlap of the enlarged system. The self
part of the shell makes the discarded Gibbs mass small for large
-$\ell$ via Markov. Perturbation replacement therefore preserves the
+$ell$ via Markov. Perturbation replacement therefore preserves the
same prediction with the base overlap on the right side.
Taylor expansion on the shell. Expansion of the fresh pattern row
fields around the base values gives a linear Gaussian field coupled
to each row of the fresh spin columns, of the form
-$n_j^{-1/2}\sum_{r\le M}h_j^r(D'_j(\xi^r))^T$. The exact denominator
+$n_j^{-1/2}sum_{rle M}h_j^r(D'_j(xi^r))^T$. The exact denominator
$√(n_j+r_j)$ may be replaced by $√(n_j)$ because $r_j$ is
-fixed. The conditional cross covariance is therefore $B_j(X,\widetilde
+fixed. The conditional cross covariance is therefore $B_j(X,widetilde
X)$ in the limit.
The deterministic second-order correction is a one-path quadratic
function $W(X,Z)$ with coefficients given by retained one-path
averages. On the shell,
-$$W(X,Z)=\frac12\sum_jr_jc_j(X)+O(ε\ell).$$
+$$W(X,Z)=frac12sum_jr_jc_j(X)+O(εell).$$
The first-order upper triangular part of $T_j-I$ has symmetric sum
$-Z_j^TZ_j/n_j$; at $Z_j^TZ_j=r_jI$ this is $-r_jI/(2n_j)$. The mean
@@ -798,33 +798,33 @@ empirical matrix overlap of the enlarged system is close to the
canonical path $q^*$.
Summing the pieces. Let $V_D(q)$ denote the limiting expected log
-increment of one independent pattern at a GG limit; the entropy $S_\theta$
-and the Gaussian functional $\mathcal G_{\theta,p}$ already entered
+increment of one independent pattern at a GG limit; the entropy $S_theta$
+and the Gaussian functional $mathcal G_calG_{theta,p}$ already entered
via the canonical lemma. The cavity comparison identifies the
-enlarged pressure with $\ell$ times the canonical value minus $o(\ell)$
+enlarged pressure with $ell$ times the canonical value minus $o(ell)$
error, so
-$$\frac1\ell\mathbb E[\log\mathcal Z_{M+\ell}-\log\mathcal Z_M]\ge\mathcal G_{\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).$$
-Uniformity in the base size. For every $\delta>0$ there is $\ell_0$
-such that for each integer $\ell\ge\ell_0$, every sequence of base
-sizes with $n_j/M\to p_j$ and increments $r_j=p_j\ell+O(1)$ satisfies
+Uniformity in the base size. For every $delta>0$ there is $ell_0$
+such that for each integer $ellgeell_0$, every sequence of base
+sizes with $n_j/Mto p_j$ and increments $r_j=p_jell+O(1)$ satisfies
-$$liminf_{M\to∞}\frac1\ell\int\Delta_{M,\ell}(u)\,d\pi(u)\ge\mathcal G_{\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
+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/M\rfloor$, divide by
-$M'$ and send $M'\to∞$, then $\ell\to∞$ and $\delta↓ 0$.
-The same parameter law $\pi$ at both endpoints makes log increments
+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
@@ -833,7 +833,7 @@ 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_{\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
integrability bound the pressure, giving convergence in expectation.
@@ -842,12 +842,12 @@ Soft penalties and quadratic tails. The Gaussian theorem was proved for smooth c
and the bounded continuous case follows by approximation. The angular formula
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 E\exp\{c|Z_j|^2\}\le C_{c,R}$ for
-$|c|$ small enough and all rows $r\le M$; the constants are uniform in
+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
@@ -855,48 +855,48 @@ partition functions tending to zero. The same comparison applies to
$V_D$ by the terminal class' explicit recursion.
Distinguished negative-quadratic. The soft-penalty Hamiltonian
-$H=\sum_rD(\xi_r)-Md\sum_\ell(\bar 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_+$:
-$$-Md\sum_\ell(\bar g_\ell-c_\ell)_+=\min_{0\le t_\ell\le d}M(t\cdot c-\sum_\ell t_\ell\bar 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).$$
The minimum is attained because the constraint set is compact and the
objective is linear in $t$. Substituting yields a Hamiltonian of the
-form $H=\sum_r(D(\xi_r)-t\cdot g(\xi_r))$ with bounded linear-$t$
+form $H=sum_r(D(xi_r)-tcdot g(xi_r))$ with bounded linear-$t$
coefficients. The unpenalised Gaussian theorem with terminal
-$D-t\cdot g$ applies uniformly in $t\in[0,d]^s$ because negative
+$D-tcdot g$ applies uniformly in $tin[0,d]^s$ because negative
quadratic coefficients remain negative, so it covers every
$t$-terminal in one stroke.
Concentration of the average. Under the natural tilted path law at
-parameter $t$, the empirical average $\bar g$ of the soft-constraint
+parameter $t$, the empirical average $barbar g$g$ of the soft-constraint
functions concentrates in probability at a deterministic vector
-$v(t)$, obtained as a limit of $\bar g$ along compactly supported
+$v(t)$, obtained as a limit of $barbar g$g$ along compactly supported
clippings. For each direction $a$ into the compact box $[0,d]^s$, the
one-sided derivative of the pressure with respect to $t$ is
-$$∂_ap(t)=-a\cdot 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 0\le \gamma\le 1,$$
+$$-(t_ell-u|a_ell|)g_ell-u|a_ell|(1-gamma)(g_ell-g_{ell,R}),qquad 0le gammale 1,$$
with nonpositive unbounded coefficients on both terms. Uniform tails
from the row-tail estimate make the directional slope error at most
-$\sum_\ell|a_\ell|ε_R$, with $ε_R\to 0$. This proves the
+$sum_ell|a_ell|ε_R$, with $ε_Rto 0$. This proves the
one-sided derivative formula, including at boundary directions.
Duality in closed form. Once the empirical average concentrates and
the derivative formula holds, standard convex duality in finite
dimensions gives the soft-penalty pressure as
-$$\min_{t\in[0,d]^s}\{t\cdot c+\mathcal G_{\theta,p}(D-t\cdot g)\}.$$
+$$min_{tin[0,d]^s}{tcdot c+mathcal G_calG_{theta,p}(D-tcdot g)}.$$
-The unpenalised functional $\mathcal G_{\theta,p}(D-t\cdot g)$ is
+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
@@ -904,91 +904,91 @@ 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)$
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 $M\times 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_1\subset E_1 ⊕ E_2\subset...$ of the desired dimensions.
+$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
+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/M\to p<1$ makes the small-ball exponent dominate the
+inequality $n/Mto p<1$ makes the small-ball exponent dominate the
net size.
./pdf/arxiv-mezzadri-0609050.pdf
-The volume Jacobian of $x\mapsto 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=\frac12\int_0^p\log(1-u)\,du.$$
+$$J_p=frac12int_0^plog(1-u),du.$$
Indeed the successive orthogonal residual lengths of $G$ are
-independent with laws $(\chi^2_{M-r+1}/M)^{1/2}$, chi-square
+independent with laws $(chi^2_{M-r+1}/M)^{1/2}$, chi-square
concentration bounds the normalised logarithms, and a Riemann sum
gives the integral.
Coefficient integral. Write $A=Udiag(s)V^T$ and put
$x=Vw$ so that $x$ is uniform on $S^N$. The column blocks of $A$
-multiply $x$ block by block: $Ax=\sum_jU_jdiag(s_j)V_j^Tx$.
+multiply $x$ block by block: $Ax=sum_jU_jdiag(s_j)V_j^Tx$.
On the Haar flag the blocks $V_j^Tx$ of a uniform coefficient decompose
-as independent uniform directions in $\mathbb R^{n_j}$ up to the
+as independent uniform directions in $mathbb R^bbR^{n_j}$ up to the
Jacobian. Thus the angular pressure satisfies
-$$P_M^f(a)=\frac1M\log\int_{S^n}e^{\sum_rf((Gx)_r)}|\det(G^TG)|^{1/2}d\sigma_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).$$
The change of measure is exact: it transfers the Haar flag weight to
the Gaussian matrix $G$ with its Jacobian. The error $o(1)$ comes from
a bounded activation and the good-norm event.
-Witnesses. Introduce auxiliary unit-block vectors $z_{i,j}\in\mathbb
-R^{n_j}$ for $j\le i<h$ and put
+Witnesses. Introduce auxiliary unit-block vectors $z_{i,j}inmathbb
+R^bbR^{n_j}$ for $jle i<h$ and put
-$$u_i=\sum_{j\le i}G_jz_{i,j}.$$
+$$u_i=sum_{jle i}G_jz_{i,j}.$$
-The test $\|y-u_i\|^2/M\le\sum_{j>i}a_j$ forces the squared norm in
+The test $|y-u_i|^2/Mlesum_{j>i}a_j$ forces the squared norm in
each tail of the orthogonal decomposition of $y$ to be no larger than
the prescribed tail, so that each initial block carries at least its
prescribed total mass. After sorting $a_j/p_j$, the weighted shell
lemma shows that these one-sided tests recover the desired weighted
shell of radii $a_j$. The witnesses are integrated as the inner
-variables of the powered Stiefel integral, with exponent $\theta\to 0$
+variables of the powered Stiefel integral, with exponent $thetato 0$
removing the entropy.
Coefficient model. With bounded $f$, the coefficient integral over
-$S^n$ decomposes into $\int\prod_jd\mu_j$ where $\mu_j$ is uniform on
+$S^n$ decomposes into $intprod_jdmu_j$ where $mu_j$ is uniform on
$S^{n_j}$, and the Gaussian column blocks $G_j$ act on each $x_j$
-independently. Encoding the constraint $\|y-u_i\|^2/M\le
-\sum_{j>i}a_j$ via the soft penalty at multiplier $t_\ell$ and sending
-the cap $\lambda/\theta$ to infinity gives exactly the functional
-$\mathcal B_{p,a}(\kappa,\lambda,B,\theta)$ after taking a supremum
+independently. Encoding the constraint $|y-u_i|^2/Mle
+sum_{j>i}a_j$ via the soft penalty at multiplier $t_ell$ and sending
+the cap $lambda/theta$ to infinity gives exactly the functional
+$mathcal B_calB_{p,a}(kappa,lambda,B,theta)$ after taking a supremum
over the trial coefficient frames $R_j$ that parametrise the inner
witness block-by-block.
The weighted shell lemma. Sort the numbers $a_j/p_j$ in nonincreasing
-order, and suppose $b_j\ge 0$ satisfy
+order, and suppose $b_jge 0$ satisfy
-$$\sum_jb_j=\sum_ja_j,\qquad \sum_{j\le i}b_j\ge\sum_{j\le i}a_j\quad(1\le i<h).$$
+$$sum_jb_j=sum_ja_j,qquad sum_{jle i}b_jgesum_{jle i}a_jquad(1le i<h).$$
Let
-$$\Phi(a)=H_p(a)+P_∞(a),\qquad H_p(a)=\frac12\sum_jp_j\log(2\pi e a_j/p_j),$$
+$$Phi(a)=H_p(a)+P_∞(a),qquad H_p(a)=frac12sum_jp_jlog(2pi e a_j/p_j),$$
-with $P_∞(a)=\lim_Mprob-P_M^f(a)$ the deterministic
-angular limit. Then $\Phi(b)\le\Phi(a)$.
+with $P_∞(a)=lim_Mprob-P_M^f(a)$ the deterministic
+angular limit. Then $Phi(b)lePhi(a)$.
Proof. Put $A$ equal to the decreasing density with values $a_j/p_j$
on the successive intervals of lengths $p_j$, and let $B^*$ equal the
@@ -1004,23 +1004,23 @@ convex-order principle. For the underlying rearrangement theory see
Hardy, Littlewood, and Polya's monograph; the specific weight-preserving
allocation proof is given.
-Consider allocation tables $\lambda_{ij}\ge 0$ with column sums $p_j$
-and row sums $p_i$. The entry $\lambda_{ij}$ is the dimension
+Consider allocation tables $lambda_{ij}ge 0$ with column sums $p_j$
+and row sums $p_i$. The entry $lambda_{ij}$ is the dimension
proportion sent from old block $j$ to new block $i$; its squared-mass
-contribution is $\lambda_{ij}b_j/p_j$. To show that some table attains
+contribution is $lambda_{ij}b_j/p_j$. To show that some table attains
$a$, fix arbitrary prices $c_i$, and let $C$ equal $c_i$ on the
interval of length $p_i$. Write $C^*$ for its decreasing rearrangement.
The support function of the attainable mass vectors assigns the largest
source densities to the largest prices. Rearrangement and then
cumulative dominance give
-$$\sum_ic_ia_i=\int_0^pCA\le\int_0^pC^*A\le\int_0^pC^*B^*=\max_\lambda\sum_{i,j}c_i\lambda_{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.$$
The second inequality is summation by parts; equal total mass permits
arbitrary, including negative, prices. Finite-dimensional separation
-gives a table attaining $a$. Set $T_{ij}=\lambda_{ij}/p_j$; then
+gives a table attaining $a$. Set $T_{ij}=lambda_{ij}/p_j$; then
-$$T\ge 0,\qquad \sum_iT_{ij}=1,\qquad Tp=p,\qquad Tb=a.$$
+$$Tge 0,qquad sum_iT_{ij}=1,qquad Tp=p,qquad Tb=a.$$
The relation is majorization relative to the block proportions $p$;
see the class of $d$-majorization polytopes studied by vom Ende and
@@ -1035,15 +1035,15 @@ 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_M\to 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
+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
+Set $Phi=-∞$ at a zero coordinate; the divergent radial entropy
and the bounded pressure $P_∞$ justify this convention in upper
bounds.
@@ -1054,53 +1054,53 @@ 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
-$\frac12\log\eta$ is below half this rate and so that
-$2\|f\|_∞\eta+\frac12\log\eta<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_{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
+$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_{\lambda\to∞}\lim_{B\to∞}\lim_{\theta↓ 0}\mathcal B_{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
+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 $2\eta$ 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 $f\in C_b(\mathbb R)$ 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
+$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_{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
@@ -1109,45 +1109,45 @@ 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
-$$A=U\,G\,V^T$$
+$$A=U,G,V^T$$
in law, with $U,V$ independent uniform orthogonal matrices, and with
$G$ acting block by block on the coefficient $x=Vw$. The spherical
coefficient $x$ decomposes into blocks of sizes $n_1,...,n_h$.
Write
-$$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),$$
-where $\rho_j\ge 0$ is the fraction of squared norm in block $j$ (with a
-null block of size $N-M$ when $\alpha<1$), $\sum_j\rho_j=1$, and the
-$\theta_j$ are unit vectors in the blocks. For a uniform coefficient
-$x$ the joint law factors: the $\theta_j$ are independent uniform unit
-vectors in the blocks, and $\rho=(\rho_j)$ is Dirichlet with
+where $rho_jge 0$ is the fraction of squared norm in block $j$ (with a
+null block of size $N-M$ when $alpha<1$), $sum_jrho_j=1$, and the
+$theta_j$ are unit vectors in the blocks. For a uniform coefficient
+$x$ the joint law factors: the $theta_j$ are independent uniform unit
+vectors in the blocks, and $rho=(rho_j)$ is Dirichlet with
parameters $(n_j/2)$ (including the null block when it is present).
Under this split, the measurement is
-$Ax=\sum_jU_j(s_j√(N\rho_j/\alpha))\,G_j\theta_j$ plus a null
-contribution. Block $j$ has image subspace $E_j=U_j\mathbb R^{n_j}$ of
-Haar dimension $n_j$ in $\mathbb R^M$, and the vector
-$y_j=s_j√(N\rho_j)G_j\theta_j$ is uniform on the sphere of radius
-$√(Ma_j)$ in $E_j$ with $a_j=s_j^2\rho_j/\alpha$. The $y_j$ are
-mutually independent once $\rho$ is fixed.
+$Ax=sum_jU_j(s_j√(Nrho_j/alpha)),G_jtheta_j$ plus a null
+contribution. Block $j$ has image subspace $E_j=U_jmathbb R^bbR^{n_j}$ of
+Haar dimension $n_j$ in $mathbb R^M$,bbR^M$, and the vector
+$y_j=s_j√(Nrho_j)G_jtheta_j$ is uniform on the sphere of radius
+$√(Ma_j)$ in $E_j$ with $a_j=s_j^2rho_j/alpha$. The $y_j$ are
+mutually independent once $rho$ is fixed.
Pressure as a Dirichlet supremum. Substitute into the pressure and use
the independence. On the coefficient sphere the factor
-$|\det(G^TG)|^{1/2}$ contributes the Jacobian rate $J_p$. The
+$|det(G^TG)|^{1/2}$ contributes the Jacobian rate $J_p$. The
normalised log of the integral over shell radii equals
-$$\sup_{\substack{\rho_i>0\\\sum_{i\in\mathcal I}\rho_i=1}}{\frac12\sum_{i\in\mathcal I}v_i\log(\rho_i)/(v_i)+\alpha P_p^f(((s_j^2\rho_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=\{1,...,h\}$ when $\alpha\ge 1$ and
-$\mathcal I=\{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
-$\frac12\sum_jp_j\log(\rho_j/p_j)$ plus a normalisation giving
-$\frac12v_j\log(\rho_j/v_j)$ after the substitution. The second term is
-the angular pressure times $\alpha$, converting per-$M$ pressure to
+$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
@@ -1157,11 +1157,11 @@ 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
+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
@@ -1170,8 +1170,8 @@ 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_N\to\alpha\in(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
+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$
+of $B$ at radius $r$. If $b=|f|_∞$, the sphere in $mathbb R^N$bbR^N$
decomposes as
-$$x=√(v)\,x'+u,\qquad x'\in\mathbb R^M,\quad u\in\mathbb R^k,$$
+$$x=√(v),x'+u,qquad x'inmathbb R^M,bbR^M,quad uinmathbb R^k,$$bbR^k,$$
-with $v=\|x\|^2/N\simBeta(M/2,k/2)$ and $x'$ uniform on
+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
+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
-$r^2\in[\alpha s_-^2v_{\rm lo},\alpha s_+^2v_{\rm hi}]$ times a
+$r^2in[alpha s_-^2v_{rm lo},alpha s_+^2v_{rm hi}]$ times a
Jacobian. The ball integral differs from the sphere integral by
-$\frac12\log$ of an explicit Beta-type factor, which is exactly the
-Dirichlet entropy of a null block of mass $1-\alpha$. The same radial
+$frac12log$ 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
-$$\max_{r\le\min(M,N)}dist(s_{N,r},[s_-,s_+])\to 0\quad\hbox{in probability}.$$
+$$max_{rlemin(M,N)}dist(s_{N,r},[s_-,s_+])to 0quadhbox{in probability}.$$
The edge interval suffices. If a sublinear number of singular values
leak outside $[s_-,s_+]$, one or two extra Haar directions act as
fake blocks. The leaked weight is assigned to those directions, and the negative
-Dirichlet entropy $\frac12v_i\log(\rho_i/v_i)$ with $v_i=o(1)$
-absorbs the effect as $\rho_i\to 0$. When $\alpha<1$, the interval may
+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
-$F_{\alpha,\nu}(f)=\alpha f(0)$.
+$F_{alpha,nu}(f)=alpha f(0)$.
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 $\lambda_x\sqrt M$ where $\lambda_x=\|Ax\|/\sqrt M$. On the
-good-norm event, $\|A\|_{op}\le C$ and at least $\kappa N$
+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
+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 $\lambda_x\sqrt M$ is
+the sphere of radius $lambda_xsqrt M$ is
-$$\frac{\Gamma(M/2)}{\Gamma((M-k)/2)(\pi M\lambda_x^2)^{k/2}}(1-(|z|^2)/(M\lambda_x^2))_+^{(M-k-2)/2}.$$
+$$frac{Gamma(M/2)}{Gamma((M-k)/2)(pi Mlambda_x^2)^{k/2}}(1-(|z|^2)/(Mlambda_x^2))_+^{(M-k-2)/2}.$$
-For $k\le M/2$ and $\lambda_x\ge\lambda_0$, this is at most
-$(C_0/\lambda_0)^k$. If a measurable set $B$ has Lebesgue measure at
-most $\delta$, then with $k=\lceil\gamma M\rceil$ 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 $0<gamma<1/2$,
-$$\mathbb P\{\#\{r:(Ax)_r\in B\}\ge k\}\le\binom{M}{k}(C_0\delta/\lambda_0)^k.$$
+$$mathbb PbbP{#{r:(Ax)_rin B}ge k}lebinom{M}{k}(C_0delta/lambda_0)^k.$$
-Choosing $R$ with $Λ^2/R^2\le\gamma$ 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 $ε+4b\gamma$, 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
-$\nu\ne\delta_0$. The proof uses the sphere-of-radius presentation and
-the small-field estimate. Set $b=\|f\|_∞$. Since
-$\nu\ne\delta_0$, the good-spectrum event has
-$\|A\|_{op}\le C$ and at least $\kappa N$ input directions
-with $s\ge s_*>0$. 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
+$nunedelta_0$. 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
-$\{z\ne 0\}$ 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
-$\sqrt\alpha$ times a Haar orthonormal $M\times N$ 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
-$f(z)=-c\min(z^2,1)$ with $c\ge 1$. On even $N$,
-$A=\sqrt\alpha U$ with $U$ a Haar orthonormal frame. For a unit input
-$u$, $\sum_rf((Au)_r)=-c\sum_r\min(M(Uu)_r^2,1)$, and a Gaussian tail
+$f(z)=-cmin(z^2,1)$ 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
-$\inf_u(1/M)\sum_r\min(M(Uu)_r^2,1)\ge 1/80$ with probability at least
-$1-e^{-c_0M}$. So the even-$N$ pressure is at most $-c\alpha/80$,
-which is $-12.5c$ for $\alpha=1000$. On odd $N$ one input direction
+$inf_u(1/M)sum_rmin(M(Uu)_r^2,1)ge 1/80$ 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
-$V\simBeta((N-1)/2,1/2)$. The event $V\le v$ has
-log-probability per $N$ tending to $\frac12\log v$, and on this event
+$sqrt Vu_{rm live}+√(1-V)u_{rm null}$ 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
-$${\rm pressure}\ge \tfrac12\log v -cv = -0.5\log 160-6.25c\approx -2.5-6.25c.$$
+$${rm pressure}ge tfrac12log v -cv = -0.5log 160-6.25capprox -2.5-6.25c.$$
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
singular measure excluding the structural zero still converges to
-$\delta_{\sqrt\alpha}$, 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
-$\nu=\sum_{k\ge 1}2^{-k}\delta_{d_k}$ with unbounded support. For
-large $N$ choose $K=\lfloor\log_2\log N\rfloor$ and multiplicities
-$n_j=\lfloor N2^{-j}\rfloor$, with the top block about $N/2$. Let $f$
-be continuous on $[0,1]$ equal to $c_K\in\{0,1\}$ on
+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
+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
-$c_K\in\{0,1\}$. Weak convergence to $\nu$ plus the outlier bound is
-not enough when $\nu$ has unbounded support; compact support is
+$c_Kin{0,1}$. 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 $s_N\to 0$ and
-$O_N$ Haar orthogonal. Let $f(z)=1_{\{z\ne 0\}}$. On even $N$ the
+$A_N=0$ 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.
+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 $Ω_1^2\in[N/4,N/2]$ has probability at least
-$\frac18 2^{-(N-3)/2}$ and on it $Ω_1^8\ge N^4/256$, so
+$N$, $|A_N|_{op}=N^{-1/4}$ 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
-$$\sum_r(A_Nx)_r^8\ge N^2/256,$$
+$$sum_r(A_Nx)_r^8ge N^2/256,$$
and therefore
-$${\rm pressure}\ge \frac1N[(N^2)/(256)+\log(\frac18 2^{-(N-3)/2})]=\frac N{256}-\frac12\log 2-(3\log 2)/(2N)\to+∞.$$
+$${rm pressure}ge frac1N[(N^2)/(256)+log(frac18 2^{-(N-3)/2})]=frac N{256}-frac12log 2-(3log 2)/(2N)to+∞.$$
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
+spectra converge to $delta_0$ with no outliers. Continuous activation
without a growth bound diverges.