diff d0ef47e..b34550f spherical_perceptron_bi_orthogonal_v2.txt
diff --git a/spherical_perceptron_bi_orthogonal_v2.txt b/spherical_perceptron_bi_orthogonal_v2.txt
index 8b6dbb3..8152060 100644
--- a/spherical_perceptron_bi_orthogonal_v2.txt
+++ b/spherical_perceptron_bi_orthogonal_v2.txt
@@ -3,7 +3,7 @@ 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
log-partition function of the spherical perceptron is
-$$F_N(A,f)=\frac1N\log\int_{S^N}\exp\left\{\sum_{r=1}^M f((Ax)_r)\right\}d\sigma_N(x).$$
+$$F_N(A,f)=\frac1N\log\int_{S^N}\exp{\sum_{r=1}^M f((Ax)_r)}d\sigma_N(x).$$
The measurement matrix $A$ carries a disorder distribution on real
$M\times N$ matrices. A temperature parameter enters by writing
@@ -134,11 +134,11 @@ 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 $s_{N,1}\ge\cdots\ge s_{N,r_N}$ for the singular
+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 $\mathrm{supp}\,\nu$ denote the support of a probability
-measure $\nu$ on $[0,\infty)$.
+$a$, and let $supp\,\nu$ denote the support of a probability
+measure $\nu$ on $[0,∞)$.
Fix $\alpha>0$, $M=\lfloor\alpha N\rfloor$, and $f\in C_b(\mathbb R)$.
Suppose the law of $A_N$ is unchanged under left and right orthogonal
@@ -147,7 +147,7 @@ uniform orthogonal $U\in O(M)$, $V\in O(N)$, and suppose the empirical
singular-value measure $r_N^{-1}\sum_{r\le r_N}\delta_{s_{N,r}}$
converges weakly in probability to a deterministic compactly supported
probability measure $\nu$. Suppose in addition that
-$\max_{r\le r_N}\mathrm{dist}(s_{N,r},\mathrm{supp}\,\nu)\to 0$ in
+$\max_{r\le r_N}dist(s_{N,r},supp\,\nu)\to 0$ in
probability. Then $F_N(A_N,f)$ converges in probability and in mean to
a deterministic value $F_{\alpha,\nu}(f)$.
@@ -158,16 +158,16 @@ 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 $s_1,\ldots,s_h$ with multiplicities $n_j$ satisfying
+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 $\mathcal I=\{1,\ldots,h\}$ when $\alpha\ge 1$ and
-$\mathcal I=\{0,1,\ldots,h\}$ when $\alpha<1$. For each block
+$v_0=1-\alpha$. 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
value for this finite approximation is
-$$\sup_{\rho}\left\{\frac12\sum_{i\in\mathcal I}v_i\log\frac{\rho_i}{v_i}+\alpha P_p^f\left(\left(\frac{s_j^2\rho_j}{\alpha}\right)_{j=1}^h\right)\right\},$$
+$$\sup_{\rho}{\frac12\sum_{i\in\mathcal I}v_i\log(\rho_i)/(v_i)+\alpha P_p^f(((s_j^2\rho_j)/(\alpha))_{j=1}^h)},$$
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
@@ -179,26 +179,26 @@ 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.
-Let $E_1,\ldots,E_h\subset\mathbb R^M$ be mutually orthogonal Haar
+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
$Ma_j$, with the point mass at $y=0$ when $a_j=0$. Define the random
angular pressure
-$$P_M^f(a)=\frac1M\log\int\exp\left\{\sum_{r=1}^M f\left(\sum_jy_{j,r}\right)\right\}\prod_j d\sigma_{E_j,Ma_j}(y_j).$$
+$$P_M^f(a)=\frac1M\log\int\exp{\sum_{r=1}^M f(\sum_jy_{j,r})}\prod_j d\sigma_{E_j,Ma_j}(y_j).$$
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
-$a\in[0,\infty)^h$, to a deterministic continuous function $P_p^f(a)$.
+$a\in[0,∞)^h$, to a deterministic continuous function $P_p^f(a)$.
For $p<1$ and positive $a$, the limit equals
-$$P_p^f(a)=J_p-H_p(a)+\lim_{\lambda\to\infty}\lim_{B\to\infty}\lim_{\theta\downarrow 0}\mathcal B_{p,a}(\kappa,\lambda,B,\theta),$$
+$$P_p^f(a)=J_p-H_p(a)+\lim_{\lambda\to∞}\lim_{B\to∞}\lim_{\theta↓ 0}\mathcal B_{p,a}(\kappa,\lambda,B,\theta),$$
for every sufficiently large fixed $\kappa$, with
-$$s_a=\sum_ja_j,\qquad H_p(a)=\frac12\sum_jp_j\log\frac{2\pi e a_j}{p_j},\qquad J_p=\frac12\int_0^p\log(1-u)\,du.$$
+$$s_a=\sum_ja_j,\qquad H_p(a)=\frac12\sum_jp_j\log(2\pi e a_j)/(p_j),\qquad J_p=\frac12\int_0^p\log(1-u)\,du.$$
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
@@ -206,7 +206,7 @@ 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 $\sum_jp_j=1$,
-$P_p^f(a)=\lim_{\eta\downarrow 0}P_{(1-\eta)p}^f(a)$ with $a$ fixed.
+$P_p^f(a)=\lim_{\eta↓ 0}P_{(1-\eta)p}^f(a)$ 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
@@ -222,22 +222,22 @@ $$g_1=y^2,\ c_1=s_a;\qquad g_2=-\min(y^2,B),\ c_2=-s_a;\qquad g_{i+2}=(y-u_i)^2,
Define
-$$\mathcal B_{p,a}(\kappa,\lambda,B,\theta)=\sup_{R_1,\ldots,R_h}\left[\sum_jp_j\left(\log\left(\kappa\sqrt{\frac{2\pi e}{p_j}}\right)+\log\frac{R_{j,11}}\kappa\right)+\theta\sum_{\ell=2}^{d_j}p_j\log\frac{R_{j,\ell\ell}}\kappa+\theta\min_{0\le t_\ell\le\lambda/\theta}\left\{t\cdot c+\mathcal G_{\theta,p}\left(\frac{f(y)}{\theta}-t\cdot g\right)\right\}\right],$$
+$$\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)}],$$
where $\mathcal G_{\theta,p}$ is the Gaussian functional obtained from
the powered Stiefel integral. The first column of
$R_j$ has weight $p_j$ and fires the outer entropy. The remaining
columns have weight $p_j\theta$ and the entropy of those columns
-vanishes as $\theta\downarrow 0$. The min over $t$ is a Lagrangian dual for the
-row-average constraints $M^{-1}\sum_r g_\ell(y_r,\dots)\le c_\ell$ in
-the limit $\lambda\to\infty$. The truncation $B$ is removed last.
+vanishes as $\theta↓ 0$. The min over $t$ is a Lagrangian dual for the
+row-average constraints $M^{-1}\sum_r g_\ell(y_r,...)\le c_\ell$ in
+the limit $\lambda\to∞$. The truncation $B$ is removed last.
The auxiliary Gaussian model distinguishes the vector whose weight is
sought from the vectors that witness its projection constraints. Fix
arbitrary column counts $d_j\ge 1$ and a positive exponent
$0<\theta<1$. Let $n_j/M\to p_j>0$ and let
-$$X_j\in\mathrm{St}(n_j,d_j):=\{X\in\mathbb R^{n_j\times d_j}:X^TX=n_jI_{d_j}\}$$
+$$X_j\inSt(n_j,d_j):=\{X\in\mathbb R^{n_j\times d_j}:X^TX=n_jI_{d_j}\}$$
be uniform on the Stiefel manifold. The first columns, jointly across
species, are outer variables; the remaining columns have the
@@ -245,11 +245,11 @@ conditional Stiefel law and are inner variables. For independent
standard Gaussian vectors $g_{r,j}\in\mathbb R^{n_j}$, define the row
fields
-$$\xi_{r,j}(X)=\frac{X_j^Tg_{r,j}}{\sqrt{n_j}},\qquad 1\le r\le M.$$
+$$\xi_{r,j}(X)=\frac{X_j^Tg_{r,j}}{√(n_j)},\qquad 1\le r\le M.$$
For a Hamiltonian $H$ define
-$$\mathcal Z_{M,\theta}(H)=\left[\int\left(\int e^H\,d\mu_{\mathrm{in}}\right)^\theta d\mu_{\mathrm{out}}\right]^{1/\theta}.$$
+$$\mathcal Z_{M,\theta}(H)=[\int(\int e^H\,d\mu_{in})^\theta d\mu_{out}]^{1/\theta}.$$
The natural tilted path law samples the outer variables with density
proportional to the inner partition function to power $\theta$, and then
@@ -265,11 +265,11 @@ normalized log integral.
Choose a common finite grid
-$$0=\zeta_0<\zeta_1<\cdots<\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 0\le i<k,$$
containing $\theta$. A trial in block $j$ consists of symmetric matrices
-$$0\preceq q_{j,0}\preceq\cdots\preceq q_{j,k-1}\preceq I_{d_j},\qquad q_{j,-1}=0,\quad q_{j,k}=I_{d_j},$$
+$$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},$$
with the pinning condition $q_{j,i}e_1=e_1$ whenever $\zeta_i\ge\theta$.
Write $q_j(u)=q_{j,i}$ on $[\zeta_i,\zeta_{i+1})$. Let
@@ -285,24 +285,24 @@ complement. Let $q_j^0$ be zero on active coordinates and the identity
on shared coordinates, with $q_{j,k}^0=I$, and define
$\mathsf K_{j,i}^0$ analogously. The block entropy is
-$$2S_\theta(q_j)=\mathrm{tr}(\mathsf P_{j,0}q_{j,0})+\sum_{i=1}^k\frac1{\zeta_i}\log\left(\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}\right),$$
+$$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}),$$
where each determinant is taken on the active space at its own index.
The normalisation compensates for the dimension change at $\theta$.
For a continuous terminal $D:\prod_j\mathbb R^{d_j}\to\mathbb R$
-satisfying $-C(1+|z|^2)\le D(z)\le C$ for some $C<\infty$, let
-$Z_0,\ldots,Z_k$ be independent centred Gaussian vectors with
+satisfying $-C(1+|z|^2)\le D(z)\le C$ for some $C<∞$, let
+$Z_0,...,Z_k$ be independent centred Gaussian vectors with
block-diagonal covariances
-$\mathrm{diag}_j(q_{j,i}-q_{j,i-1})$. Set $U_k=D$ and define backwards,
-for $i=k,k-1,\ldots,1$,
+$diag_j(q_{j,i}-q_{j,i-1})$. Set $U_k=D$ and define backwards,
+for $i=k,k-1,...,1$,
$$U_{i-1}(z)=\frac1{\zeta_i}\log\mathbb E\exp\{\zeta_iU_i(z+Z_i)\}.$$
The single-pattern value is $V_D(q)=\mathbb EU_0(Z_0)$; the outer
expectation averages the root Gaussian data. Define
-$$\mathcal G_{\theta,p}(D)=\inf_q\left\{\sum_{j=1}^hp_jS_\theta(q_j)+V_D(q)\right\},$$
+$$\mathcal G_{\theta,p}(D)=\inf_q{\sum_{j=1}^hp_jS_\theta(q_j)+V_D(q)},$$
over all eligible finite grids and trials. Refinements use a common
grid for all blocks. The diagonal increment $I-q_{j,k-1}$ is integrated
@@ -314,7 +314,7 @@ $0<\theta<1$, $n_j/M\to p_j>0$, and $D$ bounded continuous or the sum
of a bounded continuous function and a nonpositive quadratic form, the
normalized log integral satisfies
-$$\frac1M\log\mathcal Z_{M,\theta}\left(\sum_{r=1}^MD(\xi_r)\right)\longrightarrow\mathcal G_{\theta,p}(D)$$
+$$\frac1M\log\mathcal Z_{M,\theta}(\sum_{r=1}^MD(\xi_r))→\mathcal G_{\theta,p}(D)$$
in probability and in expectation. More generally, let each $g_\ell$
be bounded continuous or a nonnegative quadratic form, let
@@ -370,7 +370,7 @@ Overlaps and matrix synchronization. Write replicas as superscripts
$a,b$. The scalar and matrix overlaps of a configuration $(X,A)$ at
the disorder level are
-$$\mathcal R_j^{ab}=\frac{(X_j^a)^TX_j^b}{n_j}.$$
+$$\mathcal R_j^{ab}=((X_j^a)^TX_j^b)/(n_j).$$
The array indexed jointly by replica and column is Gram and the self
blocks are fixed at $I$. The perturbing scalarizations are
@@ -388,7 +388,7 @@ hold in the limit. Precisely, for every bounded test $F$ of the
complete arrays on the first $r$ replicas and every continuous test
$\psi$ of the generated scalarization-and-label vector on one edge,
-$$\mathbb E\langle F\psi(T^{1,r+1})\rangle=\frac1r\mathbb E\langle F\rangle\,\mathbb E\langle\psi(T^{12})\rangle+\frac1r\sum_{a=2}^r\mathbb E\langle F\psi(T^{1a})\rangle,$$
+$$\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})>,$$
where brackets denote Gibbs expectation and $\mathbb E$ averages all
randomness. Monomial GG identities give all continuous tests of this
@@ -431,7 +431,7 @@ Variation of the single-pattern value. For smooth $D$, the functional
$V_D$ is Lipschitz in the sum of the $L^1$ path distances. Along a
step variation,
-$$\left.\frac{d}{dt}V_D(q+t\dot q)\right|_{t=0}=-\frac12\sum_{j,i}w_i\,\mathrm{tr}(\dot q_{j,i}\mathbb E[M_{j,i}M_{j,i}^T]),$$
+$$\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]),$$
where $M_{j,i}$ is the conditional mean of the terminal $j$-gradient
under the successive tilted kernels. Gaussian covariance
@@ -443,23 +443,23 @@ moment matrices $G_{j,i}=\mathbb E[M_{j,i}M_{j,i}^T]$ are nondecreasing
and uniformly bounded. Put $T_{j,i}=\sum_{r=i}^{k-1}w_r(\widetilde
q_{j,r}-q_{j,r})$ and $T_{j,k}=0$. Summation by parts gives
-$$\sum_iw_i\mathrm{tr}((\widetilde q_{j,i}-q_{j,i})G_{j,i})=\mathrm{tr}(T_{j,0}G_{j,0})+\sum_{i=1}^{k-1}\mathrm{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\ \Longrightarrow\ V_D(\widetilde q)\le V_D(q).$$
+$$\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).$$
The identity and the last implication are used in the upper bound.
The canonical lemma. Consider one block, suppressing its index. Fix a
-grid and covariance levels $0\preceq b_0\preceq\cdots \preceq b_{k-1}$
+grid and covariance levels $0\preceq b_0\preceq... \preceq b_{k-1}$
of a Gaussian linear field, distinct from the spin-covariance levels
$q$. Let $\ell_n(b)$ denote the expected cascade log integral per
spatial coordinate for the Stiefel model and this linear field, after
-subtracting $(1/2)\mathrm{tr}\,b_{k-1}$. Then
+subtracting $(1/2)tr\,b_{k-1}$. Then
-$$\limsup_{n\to\infty}\ell_n(b)\le S_\theta(q)-\frac12\sum_iw_i\mathrm{tr}(q_ib_i)$$
+$$limsup_{n\to∞}\ell_n(b)\le S_\theta(q)-\frac12\sum_iw_itr(q_ib_i)$$
for every eligible $q$, locally uniformly in the field levels.
@@ -467,33 +467,33 @@ The proof identifies the right side as the value of a dual problem
whose minimiser is constructed by Gaussian tilting. Let $C=C(b)$ range
over symmetric $d\times d$ multipliers. Define
-$$\Psi(C,b)=\frac12\mathrm{tr}(Cb_0^{\phantom{.}})+\hbox{(expected tilted energy functionals)}$$
+$$\Psi(C,b)=\frac12tr(Cb_0^{\phantom{.}})+\hbox{(expected tilted energy functionals)}$$
for the tilted spin density $N(0,Q)$ and the tilted kernels attached
to $C$. The exact expression is not needed; only its structure
matters. This $\Psi(C,b)$ is lower semicontinuous in $C$ with value
-$+\infty$ outside its finite domain, and has a coercivity bound.
+$+∞$ outside its finite domain, and has a coercivity bound.
Indeed for any covariance $Q$ with $\tfrac12I\preceq Q\preceq\tfrac32 I$,
use $N(0,Q)$ as trial in the spin integration leaving the field
kernels unchanged. Write $\gamma_r=N(0,I_r)$ and $Q_1,Q_{F\mid1}$ for
the marginal and conditional laws of this trial. The chain rule for
relative entropy gives
-$$\theta^{-1}\mathrm{Ent}(Q_1\mid\gamma_1)+\mathbb E_{Q_1}\mathrm{Ent}(Q_{F\mid1}\mid\gamma_{d-1})\le\theta^{-1}\mathrm{Ent}(N(0,Q)\mid\gamma_d),$$
+$$\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),$$
and this permits correlated trial coordinates. Hence
-$$\Psi(C,b)-\tfrac12\mathrm{tr}\,C\ge\tfrac12\mathrm{tr}(C(Q-I))-\frac1{2\theta}(\mathrm{tr}\,Q-\log\det Q-d).$$
+$$\Psi(C,b)-\tfrac12tr\,C\ge\tfrac12tr(C(Q-I))-\frac1{2\theta}(tr\,Q-\log\det Q-d).$$
-Choosing $Q=I+\tfrac12\mathrm{sign}(C)$ yields
+Choosing $Q=I+\tfrac12sign(C)$ yields
-$$\Psi(C,b)-\tfrac12\mathrm{tr}\,C\ge\tfrac14\|C\|_*-c_d/\theta,$$
+$$\Psi(C,b)-\tfrac12tr\,C\ge\tfrac14\|C\|_*-c_d/\theta,$$
with $\|\cdot\|_*$ the trace norm. When $\|b_{k-1}\|\le B_0$, the
comparison constant $C=-(B_0+1)I$ gives $\Psi(C,b)\le 0$. Nested
exponents dominate $1$ and Jensen's bound controls the ordinary
Gaussian exponential integral. Thus minimisers of
-$\Psi(C,b)-\mathrm{tr}\,C/2$ lie in a compact set depending only on
+$\Psi(C,b)-tr\,C/2$ lie in a compact set depending only on
$d,\theta,B_0$. Lower semicontinuity gives existence; finiteness
forces strict precision inequalities, and differentiating in $C$
gives the tilted self-covariance identity $\mathbb Ess^T=I$.
@@ -505,7 +505,7 @@ coordinates when $\zeta_i\ge\theta$, and let $M_i=\mathbb E[s\mid
field argument, supported on the active space $F_i$. Differentiating a
single log-exponential integration gives the recursion
-$$\mathsf K_{i-1}^*=\mathsf K_i^*+\zeta_i\mathrm{Cov}(M_i\mid\mathcal H_{i-1}),$$
+$$\mathsf K_{i-1}^*=\mathsf K_i^*+\zeta_iCov(M_i\mid\mathcal H_{i-1}),$$
with deterministic conditional covariances. Averaging and the
martingale property yield
@@ -516,9 +516,9 @@ The inverse identity $\mathsf K_i\mathsf P_i\mathsf K_i=\mathsf K_i$
on the active space, combined with the pinning across $\theta$,
identifies $q_0^*$ with the precision condition. The final statement
is the canonical duality: the minimiser $C(b)$ of
-$\Psi(C,b)-\mathrm{tr}\,C/2$ yields
+$\Psi(C,b)-tr\,C/2$ yields
-$$J(b):=\Psi(C(b),b)-\tfrac12\mathrm{tr}(C(b)+b_{k-1})=S_\theta(q^*)-\frac12\sum_iw_i\mathrm{tr}(q_i^*b_i)=\inf_q\left\{S_\theta(q)-\frac12\sum_iw_i\mathrm{tr}(q_ib_i)\right\},$$
+$$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)},$$
and $q^*$ is the unique minimiser of the right side. The path $q^*$ is
eligible. For fixed $\theta,d$ and bounded $b_{k-1}$, all such
@@ -527,7 +527,7 @@ field has uniformly bounded second and fourth moments.
Uniqueness and value identification. The entropy functional
$S_\theta$ is strictly convex in the path in the
-$2S=\mathrm{tr}Pq+\cdots$ sense: expanding $A_{i-1}=A_i+\zeta_i(e_{i-1}-e_i)$
+$2S=trPq+...$ sense: expanding $A_{i-1}=A_i+\zeta_i(e_{i-1}-e_i)$
successively cancels cross terms and leaves a sum of squares. In
addition $S_\theta(q^0)=0$ and the gradient at $q^0$ vanishes; hence
$S_\theta\ge 0$. At the canonical path, $\mathsf P_0q_0^*\mathsf P_0=b_0$.
@@ -541,7 +541,7 @@ the diagonal shift inserts $\tfrac12(\mathsf K_i^*+\zeta_i\mathbb
EM_iM_i^T)$ for an increment at level $i$. Telescoping and
$\mathbb Ess^T=I$ give
-$$\dot J=-\tfrac12\sum_iw_i\mathrm{tr}(q_i^*\dot b_i),$$
+$$\dot J=-\tfrac12\sum_iw_itr(q_i^*\dot b_i),$$
and the entropy infimum has the same derivative by its unique
minimiser. Envelope differentiation is justified by the compact
@@ -552,13 +552,13 @@ Upper bound for the canonical integral. Restrict the Stiefel integral
to a small neighborhood of the identity in normalised triangular
lengths and cross readings. On that neighborhood the spin directions
have the exact Stiefel sharing law. Replacing the quadratic energy by
-$n\,\mathrm{tr}\,C/2$ costs $o_\epsilon(n)$, and replacing the
+$n\,tr\,C/2$ costs $o_ε(n)$, and replacing the
linear-field covariance by that of the exact Stiefel columns also
-costs $o_\epsilon(n)$: the normalised covariance discrepancy is
+costs $o_ε(n)$: the normalised covariance discrepancy is
uniformly small, and Gaussian interpolation bounds the log change by
-half the diagonal-minus-pair trace difference. Letting $n\to\infty$
+half the diagonal-minus-pair trace difference. Letting $n\to∞$
and then shrinking the neighborhood gives a lower bound on the
-canonical integral, so $\limsup_n\ell_n(b)\le J(b)$. The same
+canonical integral, so $limsup_n\ell_n(b)\le J(b)$. The same
covariance comparison and the compact multiplier bound give local
uniformity in $b$.
@@ -584,7 +584,7 @@ the spin columns.
Define the contact functional
-$$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_i\mathrm{tr}(\tilde q_ib_i)+\Delta(r,u),$$
+$$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),$$
where $\mathcal P_{M,r}$ is the expected (perturbed, enriched)
pressure, $\tilde q_j$ is the covariance of the linear field, $b$ is
@@ -602,7 +602,7 @@ parameters, and the field cap not binding. At this minimiser the
first-order optimality conditions act as supersolution constraints.
Differentiating in $r$ identifies the expected log-pressure of one
-fresh pattern with the negative of $r\partial A/\partial r$ at the
+fresh pattern with the negative of $r∂ A/∂ r$ at the
minimiser. The latter is bounded below by $-V_D(q^*)-\delta/2$, and
this is the upper bound on the fresh pattern log-pressure.
@@ -618,7 +618,7 @@ $q^*$ with a path of the same type.
Ghirlanda-Guerra identities. The critical step is that, at the
minimising perturbation parameters, the joint Ghirlanda-Guerra
-identities hold in the limit $M\to\infty$, and the matrix overlap
+identities hold in the limit $M\to∞$, and the matrix overlap
limits form an eligible matrix path. Supply of the GG identities uses
the monomial perturbations. Concentration of
$M^{-1}\log\mathcal Z_{M,\theta}$ on a bounded window of the
@@ -652,7 +652,7 @@ using any overlap identities. For fixed $0<\eta<1$ and trial $q$, the
path $q'=q+\eta(I-q)$ is eligible. The weighted difference against
cumulative fields is
-$$\sum_iw_i\mathrm{tr}((q_i'-q_i)b_i)=\eta\sum_r\mathrm{tr}\left(\Delta b_r\sum_{i\ge r}w_i(I-q_i)\right)\ge c_q\sum_r\mathrm{tr}\,\Delta b_r,$$
+$$\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,$$
with
@@ -708,7 +708,7 @@ $$C_a\,s\,s_M^2/M+C/(s\sqrt M)$$
for the integrated absolute fluctuation of the first derivative. At
$s=M^{-1/8}$ this implies
-$$\int\mathbb E\langle|Y_a-\mathbb E\langle Y_a\rangle|\rangle/s_M\,d\pi\to 0.$$
+$$\int\mathbb E<|Y_a-\mathbb E< Y_a>|>/s_M\,d\pi\to 0.$$
The estimates hold on a slightly larger coupling interval so endpoint
difference quotients cause no difficulty, and a diagonal selection
@@ -736,10 +736,10 @@ converging bounded outer column the conditional inner statement holds
as well. Representing a uniform frame by orthonormalised standard
Gaussian columns proves the conditional version.
-Fix small $\epsilon>0$ and restrict the enlarged partition function to
+Fix small $ε>0$ and restrict the enlarged partition function to
the thick shell
-$$\mathcal I=\{\|Z_j^TZ_j/r_j-I\|\le\epsilon\hbox{ for every }j\}.$$
+$$\mathcal I=\{\|Z_j^TZ_j/r_j-I\|\leε\hbox{ for every }j\}.$$
This only lowers its logarithm. On the shell, replace the larger
perturbation by the base perturbation at the same $u$. For fixed
@@ -759,7 +759,7 @@ Taylor expansion on the shell. Expansion of the fresh pattern row
fields around the base values gives a linear Gaussian field coupled
to each row of the fresh spin columns, of the form
$n_j^{-1/2}\sum_{r\le M}h_j^r(D'_j(\xi^r))^T$. The exact denominator
-$\sqrt{n_j+r_j}$ may be replaced by $\sqrt{n_j}$ because $r_j$ is
+$√(n_j+r_j)$ may be replaced by $√(n_j)$ because $r_j$ is
fixed. The conditional cross covariance is therefore $B_j(X,\widetilde
X)$ in the limit.
@@ -767,14 +767,14 @@ The deterministic second-order correction is a one-path quadratic
function $W(X,Z)$ with coefficients given by retained one-path
averages. On the shell,
-$$W(X,Z)=\frac12\sum_jr_jc_j(X)+O(\epsilon\ell).$$
+$$W(X,Z)=\frac12\sum_jr_jc_j(X)+O(ε\ell).$$
The first-order upper triangular part of $T_j-I$ has symmetric sum
$-Z_j^TZ_j/n_j$; at $Z_j^TZ_j=r_jI$ this is $-r_jI/(2n_j)$. The mean
fresh quadratic term is the Hessian contracted with
$Z_j^TZ_j/(2n_j)$. The diagonal specialisation is $r_jc_j(X)/2$,
and the retained matrix coefficients are bounded, so moving within
-the shell costs $O(\epsilon r_j)$. Summing over $j$ gives the
+the shell costs $O(ε r_j)$. Summing over $j$ gives the
correction. Cross-species fresh quadratic terms are centered.
Canonical prediction. Restricted to the thick shell, the Gaussian
@@ -804,13 +804,13 @@ via the canonical lemma. The cavity comparison identifies the
enlarged pressure with $\ell$ times the canonical value minus $o(\ell)$
error, so
-$$\frac1\ell\mathbb E\left[\log\mathcal Z_{M+\ell}-\log\mathcal Z_M\right]\ge\mathcal G_{\theta,p}(D)-o_{\mathrm{tol}}(1).$$
+$$\frac1\ell\mathbb E[\log\mathcal Z_{M+\ell}-\log\mathcal Z_M]\ge\mathcal G_{\theta,p}(D)-o_{tol}(1).$$
Uniformity in the base size. For every $\delta>0$ there is $\ell_0$
such that for each integer $\ell\ge\ell_0$, every sequence of base
sizes with $n_j/M\to p_j$ and increments $r_j=p_j\ell+O(1)$ satisfies
-$$\liminf_{M\to\infty}\frac1\ell\int\Delta_{M,\ell}(u)\,d\pi(u)\ge\mathcal G_{\theta,p}(D)-\delta,$$
+$$liminf_{M\to∞}\frac1\ell\int\Delta_{M,\ell}(u)\,d\pi(u)\ge\mathcal G_{\theta,p}(D)-\delta,$$
where $\Delta_{M,\ell}$ is the log-increment of the enlarged pressure
beyond the base at parameter $u$. The constant in $O(1)$ is fixed
@@ -823,7 +823,7 @@ increment bound.
Telescoping. Along the triangular chain $M'$ with $M'\equiv M$
modulo $\ell$ and block dimensions $\lfloor M'n_j/M\rfloor$, divide by
-$M'$ and send $M'\to\infty$, then $\ell\to\infty$ and $\delta\downarrow 0$.
+$M'$ and send $M'\to∞$, then $\ell\to∞$ and $\delta↓ 0$.
The same parameter law $\pi$ at both endpoints makes log increments
telescope exactly; the $o(M)$ perturbation error is removable. This
proves the matching lower bound for smooth compactly supported $D$.
@@ -845,7 +845,7 @@ sum of a bounded continuous function and a nonpositive quadratic form
$-C|\xi|^2$. Uniform tail control for Gaussian rows under the trial
path is separate from the approximation, and both are needed.
-Row tails. For every trial path $q$ and every $R<\infty$ the
+Row tails. For every trial path $q$ and every $R<∞$ the
Gaussian row fields satisfy $\mathbb E\exp\{c|Z_j|^2\}\le C_{c,R}$ for
$|c|$ small enough and all rows $r\le M$; the constants are uniform in
the finite grid and the trial path. Bounded smooth approximation of
@@ -859,7 +859,7 @@ $H=\sum_rD(\xi_r)-Md\sum_\ell(\bar g_\ell-c_\ell)_+$ is exactly the
shape in which a nonpositive quadratic enters. Rewrite the soft
penalty using the convex dual of $x_+$:
-$$-Md\sum_\ell(\bar g_\ell-c_\ell)_+=\min_{0\le t_\ell\le d}M\left(t\cdot c-\sum_\ell t_\ell\bar g_\ell\right).$$
+$$-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).$$
The minimum is attained because the constraint set is compact and the
objective is linear in $t$. Substituting yields a Hamiltonian of the
@@ -876,7 +876,7 @@ $v(t)$, obtained as a limit of $\bar g$ along compactly supported
clippings. For each direction $a$ into the compact box $[0,d]^s$, the
one-sided derivative of the pressure with respect to $t$ is
-$$\partial_ap(t)=-a\cdot v(t).$$
+$$∂_ap(t)=-a\cdot v(t).$$
Passage of slopes through clipping: interpolate only the increment in
the coupling. Increasing a nonnegative quadratic coupling adds a
@@ -887,7 +887,7 @@ $$-(t_\ell-u|a_\ell|)g_\ell-u|a_\ell|(1-\gamma)(g_\ell-g_{\ell,R}),\qquad 0\le \
with nonpositive unbounded coefficients on both terms. Uniform tails
from the row-tail estimate make the directional slope error at most
-$\sum_\ell|a_\ell|\epsilon_R$, with $\epsilon_R\to 0$. This proves the
+$\sum_\ell|a_\ell|ε_R$, with $ε_R\to 0$. This proves the
one-sided derivative formula, including at boundary directions.
Duality in closed form. Once the empirical average concentrates and
@@ -916,22 +916,22 @@ 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
orthogonalization of its consecutive column blocks of sizes
-$n_1,n_2,\ldots$ gives a Haar flag
-$E_1\subset E_1\oplus E_2\subset\cdots$ of the desired dimensions.
+$n_1,n_2,...$ gives a Haar flag
+$E_1\subset E_1 ⊕ E_2\subset...$ 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
-$C_{\mathrm{op}},C_{\mathrm{inv}}$. An elementary net argument gives
+$C_{op},C_{inv}$. 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 $(C\epsilon)^M$. Take a fixed mesh for the upper
-bound and a mesh of order $\epsilon$ for the lower bound; the strict
+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
net size.
./pdf/arxiv-mezzadri-0609050.pdf
The volume Jacobian of $x\mapsto Gx$ on the coefficient sphere is
-$\sqrt{\det(G^TG)}$. Its logarithm divided by $M$ converges in
+$√(\det(G^TG))$. Its logarithm divided by $M$ converges in
probability to
$$J_p=\frac12\int_0^p\log(1-u)\,du.$$
@@ -941,14 +941,14 @@ 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 $A=U\mathrm{diag}(s)V^T$ and put
+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: $Ax=\sum_jU_j\mathrm{diag}(s_j)V_j^Tx$.
+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
Jacobian. Thus the angular pressure satisfies
-$$P_M^f(a)=\frac1M\log\int_{S^n}e^{\sum_rf((Gx)_r)}\left|\det(G^TG)\right|^{1/2}d\sigma_n(x)+o(1).$$
+$$P_M^f(a)=\frac1M\log\int_{S^n}e^{\sum_rf((Gx)_r)}|\det(G^TG)|^{1/2}d\sigma_n(x)+o(1).$$
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
@@ -985,9 +985,9 @@ $$\sum_jb_j=\sum_ja_j,\qquad \sum_{j\le i}b_j\ge\sum_{j\le i}a_j\quad(1\le i<h).
Let
-$$\Phi(a)=H_p(a)+P_\infty(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)=\frac12\sum_jp_j\log(2\pi e a_j/p_j),$$
-with $P_\infty(a)=\lim_M\mathrm{prob}\text{-}P_M^f(a)$ the deterministic
+with $P_∞(a)=\lim_Mprob-P_M^f(a)$ the deterministic
angular limit. Then $\Phi(b)\le\Phi(a)$.
Proof. Put $A$ equal to the decreasing density with values $a_j/p_j$
@@ -1035,51 +1035,51 @@ 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 $\epsilon_M\to 0$.
+enlarged new $a$-shell with failure probability $ε_M\to 0$.
./pdf/arxiv-1911.01061v5.pdf
Angular identity. Radial integration and Stirling's formula show that
-the weighted Lebesgue volume in $E_1\oplus\cdots\oplus E_h$ of a small
+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_\infty(a)$. The
+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=-\infty$ at a zero coordinate; the divergent radial entropy
-and the bounded pressure $P_\infty$ 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
-$\epsilon$, so the integral on the $a$-shell is at least the integral
-on the $b$-shell up to $O(\epsilon)$.
+$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 $\epsilon$.
+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\|_\infty\eta+\frac12\log\eta<\epsilon$. Choose shell widths and
+$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 $\epsilon$. A
+$\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
-$\epsilon$ loss. Thresholds may depend on the fixed $\lambda$; no
+$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\downarrow 0$ because the inner expression is an increasing
+$\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\infty}\lim_{B\to\infty}\lim_{\theta\downarrow 0}\mathcal B_{p,a}(\kappa,\lambda,B,\theta).$$
+$$\Phi(a)=J_p+\lim_{\lambda\to∞}\lim_{B\to∞}\lim_{\theta↓ 0}\mathcal B_{p,a}(\kappa,\lambda,B,\theta).$$
Using the shell-volume identity for $\Phi$ proves the angular formula
-and identifies every subsequential $P_\infty$. Equicontinuity extends
+and identifies every subsequential $P_∞$. Equicontinuity extends
convergence to compact sets and to zero radii.
Full output dimension. If $\sum_jp_j=1$, retain, independently in each
@@ -1090,7 +1090,7 @@ the old Gibbs mass with probability tending to one over the chosen
retained subspaces: for each fixed old vector the projection failure
probability is uniformly small, so Fubini and Markov apply after
arbitrary bounded activation reweighting. Zero-radius blocks are
-ignored. Letting $\eta\downarrow 0$ recovers the $p=1$ case.
+ignored. Letting $\eta↓ 0$ recovers the $p=1$ case.
Continuity in $f$ and $a$. For $f\in C_b(\mathbb R)$ choose bounded
Lipschitz approximants with a common bound and uniform convergence on
@@ -1105,7 +1105,7 @@ This completes the angular theorem.
The radial decomposition. Return to the spherical integral with $A$ of
bi-orthogonally invariant law. Fix the singular-value block
-approximation $s_1,\ldots,s_h$ with multiplicities $n_j$, and let $G$
+approximation $s_1,...,s_h$ with multiplicities $n_j$, and let $G$
be the Gaussian flag matrix from the angular reduction. The
decomposition of $A$ yields
@@ -1113,10 +1113,10 @@ $$A=U\,G\,V^T$$
in law, with $U,V$ independent uniform orthogonal matrices, and with
$G$ acting block by block on the coefficient $x=Vw$. The spherical
-coefficient $x$ decomposes into blocks of sizes $n_1,\ldots,n_h$.
+coefficient $x$ decomposes into blocks of sizes $n_1,...,n_h$.
Write
-$$x=\sqrt{\rho_0N}\theta_0+\sum_j\sqrt{\rho_jN/s_j^2}\theta_j\sqrt{s_j^2/N},$$
+$$x=√(\rho_0N)\theta_0+\sum_j√(\rho_jN/s_j^2)\theta_j√(s_j^2/N),$$
where $\rho_j\ge 0$ is the fraction of squared norm in block $j$ (with a
null block of size $N-M$ when $\alpha<1$), $\sum_j\rho_j=1$, and the
@@ -1126,11 +1126,11 @@ vectors in the blocks, and $\rho=(\rho_j)$ is Dirichlet with
parameters $(n_j/2)$ (including the null block when it is present).
Under this split, the measurement is
-$Ax=\sum_jU_j(s_j\sqrt{N\rho_j/\alpha})\,G_j\theta_j$ plus a null
+$Ax=\sum_jU_j(s_j√(N\rho_j/\alpha))\,G_j\theta_j$ plus a null
contribution. Block $j$ has image subspace $E_j=U_j\mathbb R^{n_j}$ of
Haar dimension $n_j$ in $\mathbb R^M$, and the vector
-$y_j=s_j\sqrt{N\rho_j}G_j\theta_j$ is uniform on the sphere of radius
-$\sqrt{Ma_j}$ in $E_j$ with $a_j=s_j^2\rho_j/\alpha$. The $y_j$ are
+$y_j=s_j√(N\rho_j)G_j\theta_j$ is uniform on the sphere of radius
+$√(Ma_j)$ in $E_j$ with $a_j=s_j^2\rho_j/\alpha$. The $y_j$ are
mutually independent once $\rho$ is fixed.
Pressure as a Dirichlet supremum. Substitute into the pressure and use
@@ -1138,13 +1138,13 @@ the independence. On the coefficient sphere the factor
$|\det(G^TG)|^{1/2}$ contributes the Jacobian rate $J_p$. The
normalised log of the integral over shell radii equals
-$$\sup_{\substack{\rho_i>0\\\sum_{i\in\mathcal I}\rho_i=1}}\left\{\frac12\sum_{i\in\mathcal I}v_i\log\frac{\rho_i}{v_i}+\alpha P_p^f\left(\left(\frac{s_j^2\rho_j}{\alpha}\right)_{j=1}^h\right)\right\}+o(1),$$
+$$\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),$$
where $v_j=\alpha p_j$, $v_0=1-\alpha$ when $\alpha<1$, and
-$\mathcal I=\{1,\ldots,h\}$ when $\alpha\ge 1$ and
-$\mathcal I=\{0,1,\ldots,h\}$ when $\alpha<1$. The first term is the
+$\mathcal I=\{1,...,h\}$ when $\alpha\ge 1$ and
+$\mathcal I=\{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}\cdots$, contributing
+$\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
@@ -1153,7 +1153,7 @@ 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,\ldots,s_h)$ and on
+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.
@@ -1170,7 +1170,7 @@ 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\|_\infty$, so probability convergence of
+uniformly bounded by $\|f\|_∞$, so probability convergence of
$\sup$ and $\inf$ plus boundedness gives convergence of expectations.
No separate argument is needed.
@@ -1179,18 +1179,18 @@ 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,\infty)$. The angular and
+$(M_N,N_N)$ with $M_N/N_N\to\alpha\in(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\|_\infty$, the sphere in $\mathbb R^N$
+of $B$ at radius $r$. If $b=\|f\|_∞$, the sphere in $\mathbb R^N$
decomposes as
-$$x=\sqrt{v}\,x'+u,\qquad x'\in\mathbb R^M,\quad u\in\mathbb R^k,$$
+$$x=√(v)\,x'+u,\qquad x'\in\mathbb R^M,\quad u\in\mathbb R^k,$$
-with $v=\|x\|^2/N\sim\mathrm{Beta}(M/2,k/2)$ and $x'$ uniform on
-$S^{M-1}$ conditionally. The measured field at radius $\sqrt{Nv}$ has
+with $v=\|x\|^2/N\simBeta(M/2,k/2)$ and $x'$ uniform on
+$S^{M-1}$ conditionally. The measured field at radius $√(Nv)$ has
squared norm approximately $\alpha s_N^2v$ when $s_N$ is the typical
singular value. The endpoints $v$ small and $v$ near one contribute
negligibly because the Beta density makes such values exponentially rare at
@@ -1204,11 +1204,11 @@ formula holds with $M$ denominators and the extra null block providing
the endpoint.
Edge intervals and sublinear outliers. The assumption that every
-singular value stays in $\mathrm{supp}\,\nu$ weakens as follows. Let
-$s_-=\min\mathrm{supp}\,\nu$ and $s_+=\max\mathrm{supp}\,\nu$. The
+singular value stays in $supp\,\nu$ weakens as follows. Let
+$s_-=\minsupp\,\nu$ and $s_+=\maxsupp\,\nu$. The
requirement becomes
-$$\max_{r\le\min(M,N)}\mathrm{dist}(s_{N,r},[s_-,s_+])\to 0\quad\hbox{in probability}.$$
+$$\max_{r\le\min(M,N)}dist(s_{N,r},[s_-,s_+])\to 0\quad\hbox{in probability}.$$
The edge interval suffices. If a sublinear number of singular values
leak outside $[s_-,s_+]$, one or two extra Haar directions act as
@@ -1225,7 +1225,7 @@ $f$ bounded. It relaxes to $|f(z)|/(1+z^2)\to 0$ with $f$ continuous.
Conditional on input coefficients and singular values, Haar left
rotation makes the field a uniformly random vector on the sphere of
radius $\lambda_x\sqrt M$ where $\lambda_x=\|Ax\|/\sqrt M$. On the
-good-norm event, $\|A\|_{\mathrm{op}}\le C$ and at least $\kappa N$
+good-norm event, $\|A\|_{op}\le C$ and at least $\kappa N$
input directions have singular value at least $s_*>0$. For $L>0$
there is small $\lambda_0>0$ with
@@ -1235,7 +1235,7 @@ independently of the spectrum. Conditional on input coefficients and
singular values, the density of any $k<M-2$ entries of the field on
the sphere of radius $\lambda_x\sqrt M$ is
-$$\frac{\Gamma(M/2)}{\Gamma((M-k)/2)(\pi M\lambda_x^2)^{k/2}}\left(1-\frac{|z|^2}{M\lambda_x^2}\right)_+^{(M-k-2)/2}.$$
+$$\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}.$$
For $k\le M/2$ and $\lambda_x\ge\lambda_0$, this is at most
$(C_0/\lambda_0)^k$. If a measurable set $B$ has Lebesgue measure at
@@ -1243,19 +1243,19 @@ most $\delta$, then with $k=\lceil\gamma M\rceil$ and $0<\gamma<1/2$,
$$\mathbb P\{\#\{r:(Ax)_r\in B\}\ge k\}\le\binom{M}{k}(C_0\delta/\lambda_0)^k.$$
-Choosing $R$ with $\Lambda^2/R^2\le\gamma$ and approximating $f$ in
+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
the extension. The exceptional inputs have exponentially small
probability and negligible partition mass; on the remaining inputs the
-normalised energy difference is bounded by $\epsilon+4b\gamma$, which
+normalised energy difference is bounded by $ε+4b\gamma$, 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\|_\infty$. Since
+the small-field estimate. Set $b=\|f\|_∞$. Since
$\nu\ne\delta_0$, the good-spectrum event has
-$\|A\|_{\mathrm{op}}\le C$ and at least $\kappa N$ input directions
+$\|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
@@ -1284,8 +1284,8 @@ $\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
maps to zero; a uniform input decomposes as
-$\sqrt Vu_{\rm live}+\sqrt{1-V}u_{\rm null}$ with
-$V\sim\mathrm{Beta}((N-1)/2,1/2)$. The event $V\le v$ has
+$\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
the energy is at least $-cv$, giving
@@ -1329,16 +1329,16 @@ 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\|_{\mathrm{op}}=N^{-1/4}$ and
-$\sum_r(A_Nx)_r^8=N^{-2}\sum_r\Omega_r^8$ with $\Omega=O_Nx$. The
-coordinate event $\Omega_1^2\in[N/4,N/2]$ has probability at least
-$\frac18 2^{-(N-3)/2}$ and on it $\Omega_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_1^2\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
$$\sum_r(A_Nx)_r^8\ge N^2/256,$$
and therefore
-$${\rm pressure}\ge \frac1N\left[\frac{N^2}{256}+\log\left(\frac18 2^{-(N-3)/2}\right)\right]=\frac N{256}-\frac12\log 2-\frac{3\log 2}{2N}\to+\infty.$$
+$${\rm pressure}\ge \frac1N[(N^2)/(256)+\log(\frac18 2^{-(N-3)/2})]=\frac N{256}-\frac12\log 2-(3\log 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