A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

Proof. A linear change of variables reduces the assertion to S=\Id. First suppose V(x)=|x|^2/2+U(x) is smooth with bounded Hessian. The diffusion with generator L=\Delta-DV\cdot D has globally Lipschitz drift and invariant law \pi. If J_t is the derivative of its trajectory with respect to the starting point, then \dot J_t=-D^2V(X_t)J_t, whence \|J_t\|_{\op}\le e^{-t}. For smooth bounded positive q, bounded away from zero, weighted Cauchy–Schwarz gives

Integration by parts yields

Couple a trajectory started at x to an independent stationary initial point with the same driving Brownian motion. Their distance is at most e^{-t} times their initial distance. Since \pi has Gaussian tails, this proves convergence of P_tq to \E_\pi q for bounded Lipschitz q, and hence convergence of its entropy to zero. Integrating the preceding two displays gives \Ent_\pi(q)\le\frac12\int|Dq|^2/q\,d\pi. Apply this to q=f^2+\varepsilon and let \varepsilon\downarrow0.

For general U, take its quadratic infimal convolutions

The minimizer is unique, and monotonicity of the subgradient of U shows that DU_j is Lipschitz. The functions U_j are convex and increase to U. A fixed affine minorant U(z)\ge\ell\cdot z+c gives U_j(x)\ge\ell\cdot x+c-|\ell|^2/(4j). Mollify U_j with a smooth compactly supported kernel at radii tending to zero. The resulting functions are smooth, convex, and have bounded Hessian; the affine lower bound persists with a uniform additive error. Consequently their normalized densities converge almost everywhere and in L^1, dominated by an integrable Gaussian times an exponential of a linear function. The compact-test inequality passes to the limit. Boundary values of U do not matter, since the boundary of a full dimensional convex support has Lebesgue measure zero. Finally, insert f=1+\varepsilon g, using finite-energy cutoffs of the constant, and compare the coefficients of \varepsilon^2 to obtain the variance inequality.□

Proof. Integration by parts gives \langle f,A_\pi g\rangle=\int Df\cdot Dg\,d\pi. If v\in L^2(\pi) is orthogonal to (1+A_\pi)C_c^\infty, then (1+A_\pi)v=0 distributionally. Conjugating by \sqrt\pi reduces this to a Poisson equation with a smooth local potential. The local Newtonian potential estimates and bootstrap make v smooth; see [20, Lecture 24, p. 2; Lecture 16, p. 2]. Take a compact cutoff \chi_R, equal to one on the ball of radius R, with |D\chi_R|\le C/R. Testing by \chi_R^2v and using 2ab\le a^2/2+2b^2 gives

Thus v=0. The range of 1+A_\pi is dense. Nonnegativity gives \|(1+A_\pi)f\|_2\ge\|f\|_2, so the range of the closure is also closed and therefore equals L^2. The closure is self-adjoint: for g in its adjoint domain, solve (1+\overline A)h=(1+A^*)g; then g-h is in the orthogonal complement of the range of 1+A, hence vanishes. This also proves the asserted maximal distributional domain characterization.

The cutoffs \chi_R converge to 1 in the form norm with energy tending to zero. Hence 1 belongs to the nullspace of the closed form and P_u1=1. To identify the diffusion, solve the stochastic equation locally and kill it on exiting each ball. Its kernels are symmetric and sub-Markov, as follows either from the Dirichlet heat equation or integration by parts. Increasing the balls gives the minimal sub-Markov semigroup. The stopped Itô formula on compact tests identifies its generator as an extension of -A_\pi. Essential self-adjointness identifies it with P_u, which loses no mass. Started in \pi, the resulting diffusion is stationary and nonexplosive. Finally approximate f in graph norm by compact smooth functions. Their gradients converge in L^2(\pi) since \|Dg\|_2^2=\langle g,Ag\rangle. Stationarity and the Itô isometry pass every term of their formulas to the limit.□

Proof. For a bounded density q bounded away from zero, the form chain rule, first for smooth functions and then by form approximation, gives

Spectral smoothing puts P_uq in the operator domain for u>0; the bounded range permits 1+\log(P_uq) as a form test. Approximate an arbitrary finite-entropy density by truncating it above and below and normalizing. The initial entropies converge. The semigroup contracts L^1, so densities converge at every later time. Lower semicontinuity of entropy and of 4\int|D\sqrt q|^2d\pi, followed by Fatou’s lemma in time, gives Equation (2.5) from time zero. Restarting the argument gives arbitrary a.

For simultaneous heat evolution put z=p_r/q_r. Direct differentiation, using \partial_r p_r=\Delta p_r/2 and the same equation for q_r, gives

Here use smooth convex \Phi_k with \Phi_k(1)=\Phi_k'(1)=0, with second derivatives increasing to 1/z, and with second derivatives supported in compact subsets of (0,\infty). Such functions have at most linear growth and increase to z\log z-z+1. Integrate Equation (2.7) against \chi_R with |\Delta\chi_R|\le C/R^2. At each fixed k the cutoff error vanishes because q_r\Phi_k(p_r/q_r)\le C_k(p_r+q_r). Let R\to\infty and then k\to\infty. Monotone convergence gives the entropy and information terms, and in particular the stated inequality on positive time intervals. Data processing gives H(p_\varepsilon\mid q_\varepsilon)\le H(p_0\mid q_0); let \varepsilon\downarrow0 to include the initial time.

For Ornstein–Uhlenbeck evolution relative to \gamma, apply the same convex truncations to the ratio. Its generator is \Delta-x\cdot D. Expanding radial cutoffs have bounded generator, tending pointwise to zero, so the error vanishes by domination at fixed k. This gives equality on positive time intervals. At time zero, relative entropy is continuous: data processing gives its upper bound and lower semicontinuity under weak convergence gives its lower bound. Monotone convergence of the nonnegative time integral includes zero. Tonelli’s theorem proves all conditional versions.□

Proof. Conditional convexity of relative entropy and H(\gamma(\cdot-\sqrt t s)\mid\gamma)=t|s|^2/2 give the upper bound. Gaussian kernel differentiation gives

The differentiation is justified at each positive t by the Gaussian kernel and the finite second moment, or by truncating S and passing locally and in the conditional squared norm. Applying Ornstein–Uhlenbeck evolution for time u replaces t by te^{-2u}. Its entropy dissipation identity and dt/du=-2t give d'(t)=\frac12\E|\E[S\mid Z_t,U]|^2 almost everywhere on (0,\infty). The upper bound implies d(t)\to0 at zero, proving the integrated identity. For 0<v<t, the channel at v is obtained from the channel at t by multiplying its output by \sqrt{v/t} and adding fresh Gaussian noise of covariance (1-v/t)\Id. Conditional Jensen then proves monotonicity of prediction energy.□

Proof of Corollary 1.2. Jensen’s inequality in each coordinate gives \int|x|^2d\mu\le na^2\log2, so \mu\in\mathcal P_2(\R^n). Only the case H(\nu\mid\mu)<\infty needs proof. For s>0, put \gamma_s=N(0,s\Id), \mu_s=\mu*\gamma_s, and \nu_s=\nu*\gamma_s. The density of \mu_s is smooth and strictly positive. For X\sim\mu and an independent standard Gaussian G, its potential V_s satisfies

Indeed, the posterior has the original convex potential plus |x-y|^2/(2s), so Lemma 2.1 bounds its covariance by s\Id. Gaussian differentiation is valid without compact support. Thus \mu_s is centered and log-concave, with a smooth convex potential.

Set \beta_s=\sqrt{8s/3} and a_s=a+\beta_s. For a scalar standard Gaussian Z, \E\exp(sZ^2/\beta_s^2)=2. The inequality

and Hölder’s inequality show that \mu_s satisfies (1.1) with parameter a_s. Theorem 1.1, with the Sobolev extension in Section 2, gives entropy coefficient Ca_s^2. In the conventions of [17, Definitions 1–2], \Ent(f^2)\le L\int|Df|^2 is \mathrm{LSI}(2/L), whose transport conclusion is W_2^2\le LH. Their Theorem 1 applies to the smooth convex potential of \mu_s, and hence

The last inequality is data processing for the common Gaussian kernel, as in Section 2. Coupling a vector with itself plus independent Gaussian noise gives W_2(\mu_s,\mu),W_2(\nu_s,\nu)\le\sqrt{ns}. Letting s\downarrow0 at fixed n, with a_s\to a, proves the claim.□

Proof. Choose laws and compact smooth tests for which the entropy-to-energy ratio divided by the squared marginal parameter tends to infinity. The tests may be chosen with positive energy: a smooth test of zero energy is constant on the connected interior of the convex support, and its entropy vanishes. Restriction of the law to a sufficiently large ball preserves each test quotient arbitrarily closely. Multiply the restricted density by e^{-\varepsilon|x|^2/2} and normalize, with \varepsilon>0 sufficiently small to preserve that quotient again. Both operations preserve log-concavity, their combined density relative to the original law can be bounded by 2, and Lemma 2.1 gives a finite logarithmic Sobolev constant for the resulting compact law.

A bounded density increase changes the exponential-square expectation by at most that factor. If \E e^{Z^2/A^2}\le4, Jensen gives \E e^{Z^2/(2A^2)}\le2. Centering costs only another absolute factor: \E Z^2\le A^2\log4, while (Z-\E Z)^2\le2Z^2+2(\E Z)^2; increase the denominator first to make the resulting exponential expectation bounded by an absolute constant, and once more by Jensen to make it at most 2. These arguments apply separately to every unit linear form with the same constants. Translation preserves the test quotient. Thus the centered compact laws still have R/A^2\to\infty with 0<R<\infty. Dilation by R^{-1/2} gives Equation (3.1).

For completeness, if a centered scalar Z obeys \E e^{Z^2/a^2}\le2, then \E|Z|^k\le2a^k\Gamma(k/2+1), by integrating its tail. Expansion of e^{tZ}, with the linear term absent, therefore gives \log\E e^{tZ}\le Ca^2t^2 for a|t|\le c. For a|t|\ge c, use tZ\le Z^2/a^2+a^2t^2/4 and absorb \log2 into Ca^2t^2. Apply this to Z=X\cdot\theta. The second moment bound follows either from the exponential-square bound or by differentiating the centered moment generating function at zero.□

Proof. The heat equation for \mu_s is the continuity equation with velocity DV_s/2. On a compact positive-time interval this velocity is globally Lipschitz and grows at most linearly, so its smooth flow is a diffeomorphism carrying \mu_s to \mu_t. Its Jacobian satisfies \dot J=K_tJ/2. By positivity of K_t, the flow expands distances and its inverse is 1-Lipschitz when t\ge s. Transport a test by that inverse. Its entropy is unchanged and its Dirichlet energy cannot increase, proving R_t\ge R_s. Equation (2.2) gives R_{s+h}\le R_s+h and R_s\le1+s. For every compact smooth test, its entropy and energy converge to those under \mu as s\downarrow0. Taking the supremum of its quotient gives \liminf_{s\downarrow0}R_s\ge R(\mu)=1. Monotonicity now proves the remaining bound.□

Proof. Write r=d\nu/d\gamma_s. The Gaussian kernel formula gives D\log r=b_s/s, which is bounded, and hence |\log r(y)|\le C(1+|y|). For v=u\sqrt r,

Gaussian integration by parts followed by Cauchy–Schwarz gives

Initially this calculation is for compact tests, and cutoffs extend it by lower semicontinuity. The linear-growth entropy correction is consequently bounded by \delta\|Dv\|_2^2+C_\delta\|v\|_2^2 for every \delta>0. Gaussian logarithmic Sobolev and the displayed derivative formula, with another arbitrarily small Cauchy–Schwarz parameter, prove Equation (3.6). The same moment estimate shows tightness of the L^2(\nu) mass of any bounded form sequence. On a fixed ball, the smooth density is bounded above and away from zero, so the ordinary local Rellich theorem gives L^2 precompactness. A diagonal extraction and tightness prove compactness globally.

We give the tail argument for the sharp coefficient 2s in Equation (3.7). Put v_{j,L}=(|u_j|-L)_+ and m_{j,L}=\|v_{j,L}\|_2^2. Strong L^2 convergence implies \lim_{L\to\infty}\sup_jm_{j,L}=0. The energies of v_{j,L} and |u_j|\wedge L sum to \|Du_j\|_2^2. Lower semicontinuity for the clipped functions therefore gives

For a nonnegative v of squared norm m\le1, the negative part of v^2\log v^2 has integral at most C\sqrt m (use -z\log z\le C\sqrt z on 0\le z\le1). Since m\log m\le0, Equation (3.6) thus gives

If |u_j|>ML, where M>1 is fixed, then |u_j|\le c_M v_{j,L} with c_M=M/(M-1). Consequently

The integrals of bounded truncations of z^2\log z^2 pass to the limit by strong L^2 convergence; its negative part does as well, being bounded and continuous. The last two bounds and Equation (3.8), with limits in the order j, L, M, \varepsilon, show that the lost positive entropy is at most 2s(E_*-\|Du\|_2^2). The normalization terms are zero because \|u\|_2=\|u_j\|_2=1.□

Proof. Take nonnegative unit-norm maximizing tests f_j. Replacing a test by its absolute value preserves entropy and energy, so this choice loses nothing. Equation (3.6), with s+\varepsilon<R, bounds their energies. We first exclude energies tending to zero; this is the only place the strict inequality R>P is required.

Suppose \varepsilon_j^2=\|Df_j\|_2^2\to0, and write c_j=\E_\nu f_j and h_j=(f_j-c_j)/\varepsilon_j. Poincaré gives \|h_j\|_2^2\le P, while \|Dh_j\|_2^2=1 and 1-c_j^2=\varepsilon_j^2\|h_j\|_2^2. Thus c_j\to1. By compactness, after extraction, h_j\to h in L^2, where

For c>0, \varepsilon>0 and z\ge-c/\varepsilon define

Normalization and -\log c_j^2-1+c_j^2\ge0 give

Here is the scalar estimate needed to control unbounded h_j. For 1/2\le c\le1 and 0<\varepsilon\le1/2,

uniformly on bounded z-intervals as c\to1 and \varepsilon\to0. To verify the bound, put t=\varepsilon z/c. Then Q=z^2B(t), where Taylor’s integral formula gives

For t\le0, B(t)\le2. For t\ge0, use t\le z to obtain B(t)\le2+2\log(1+z)\le2+2\log2+\log_+z^2. Continuity at t=0 proves the compact convergence.

The tail proof in Lemma 3.3 did not use unit norm until its last sentence. In particular, for any bounded form sequence u_j\to u in L^2 with energies tending to E_*, it proves

Apply the compact convergence in Equation (3.11) on |h_j|\le K and its upper bound on the complement. Strong L^2 convergence removes the quadratic tail as K\to\infty; Equation (3.12) controls the entropy tail. Equation (3.10) consequently gives

This contradicts the maximizing property.

We may therefore extract f_j\to f strongly in L^2 with \|Df_j\|_2^2\to E_*>0. Put E=\|Df\|_2^2. The entropy defect and the sharp inequality give

Since R>s and E\le E_*, equality forces E=E_*>0. Thus f is a nonnegative, nonconstant optimizer.

The homogeneous functional \Ent(g^2)-2R\|Dg\|_2^2 is nonpositive and is zero at f. Its derivative along every compact smooth variation \zeta is

The entropy normalization term cancels the additional linear term. Differentiation is legitimate also at zeros because z^2\log z^2 has derivative zero at zero. This proves the Euler equation in Equation (3.9).

We have obtained a nonnegative optimizer with positive energy. To use its logarithmic derivative, we still need local regularity and strict positivity. We spell out the weak-solution regularity step, using the interior Newtonian-potential L^p and Schauder estimates; see [20, Lecture 24, p. 2; Lecture 10, Proposition 1]. Write \nu=e^{-V} and set v=e^{-V/2}f\ge0. The function V is smooth, and direct differentiation of the weak equation gives

The conjugating factor and its reciprocal are bounded on each fixed ball. Since |z\log z|\le C_\delta(1+z^{1+\delta}) for z\ge0, local L^p integrability of v puts the right side in L^q_{\rm loc}, where q=p/(1+\delta)>1. Localize that right side on a larger ball and take its Newtonian potential with the sign solving -\Delta u=F. The cited L^q estimate gives u\in W^{2,q}_{\rm loc}; the difference v-u is distributionally harmonic and hence smooth by Weyl’s lemma [20, Lecture 3, Theorem 2]. This proves the required W^{2,q} gain directly for the weak solution.

For n\ge2, start with p=2 and choose 0<\delta<\min(1,2/n). Before boundedness is reached, Sobolev embedding improves reciprocal integrability from 1/p to (1+\delta)/p-2/n, a decrease greater than 1/n because p\ge2. At a critical exponent use any sufficiently large finite integrability exponent and continue once more. After finitely many steps q>n/2, so v, and thus f, is locally bounded and Hölder continuous. In dimension one the same conclusion follows by twice integrating the local Poisson equation. The function z\log z is locally Hölder of every exponent below one on [0,\infty); reducing the exponent if necessary, the Schauder estimate therefore gives f\in C^{2,\alpha}_{\rm loc} for some \alpha>0. All constants in this paragraph concern one fixed law and dimension; none enters a uniform estimate along the counterexample sequence.

If f(x_0)=0, choose a ball B about x_0 with 0\le f\le b<1 on its closure. Start the \nu-diffusion at x_0 and stop at its exit time \tau from B. For Z_t=f(Y_{t\wedge\tau}), m(t)=\E Z_t, and g(z)=-\lambda z\log z, the stopped Itô formula gives, for almost every t,

Indeed g is nonnegative and concave on [0,b]; the first inequality adds its nonnegative stopped boundary contribution. Also m'\ge0. If m(t_1)>0, let t_a be the first time at which m=a, for 0<a<m(t_1). Integration gives

contradicting divergence as a\downarrow0. Hence m=0. The killed nondegenerate diffusion has positive probability of reaching each open subset of B while staying in B: one can follow a smooth interior path by requiring Brownian motion to remain in a small tube about its corresponding driving path, and use local Lipschitz continuity of the drift. Brownian tube probabilities are positive by its finite increment densities and bridge continuity. Consequently f vanishes throughout B. Its zero set is both open and closed, impossible in connected \R^n for a unit-norm function. Thus f>0, and smooth elliptic bootstrap makes f smooth everywhere.□

Proof. Since w=2Df/f, I=4\|Df\|_{L^2(\nu)}^2; optimizer equality gives I=2\lambda H(\eta\mid\nu)>0. The chain rule applied to its Euler equation gives the first identity in Equation (3.13). Commuting one derivative with A_\nu gives

Since A_\eta=A_\nu-w\cdot D and Dw is symmetric, the second identity follows distributionally. Its right side is in L^2(\eta), since K is bounded and w\in L^2(\eta). Lemma 2.2 therefore puts w in the operator domain; no unproved integrability of higher derivatives has been used.

Let p=g^2\eta have unit mass, where g is compactly supported and smooth. Direct expansion and compactly supported integration by parts give

These formulas may be read in square-root form at zeros of g. On the other hand H(p\mid\nu)=H(p\mid\eta)+\int h\,dp. Thus

Apply the inequality for \nu to the compact smooth function gf. Its deficit is nonnegative, so the right side is nonnegative. Homogeneity and the Sobolev approximation of Section 2 prove R(\eta)\le1/\lambda. Finally the domain identity gives

while Poincaré for \eta gives the lower bound \lambda(I-|\E_\eta w|^2). Subtract to obtain Equation (3.14).□

Proof. The centered linear moment generating functions under \mu_s obey

The first follows by independent Gaussian addition and Equation (3.2); the second follows by conditional Jensen from that equation. Entropy duality, optimized over the scalar multiple of each unit vector, gives

All expectations exist. More explicitly, compact support of X makes \E_{\mu_s}e^{c|Y|^2}<\infty for some c>0; entropy duality then implies \E_\sigma|Y|^2<\infty. The latter fact is used only for integrability, not for a uniform bound in dimension.

Let \sigma_u be the translate of \sigma by -up, 0\le u\le1, and let D_u=H(\sigma_u\mid\mu_s). Change of variables in entropy and the Hessian bound give

Thus Equation (3.17), applied to \sigma_u, implies |\E_\sigma b_s(Y-up)|\le C_Ja\sqrt{D_0} uniformly in u. Differentiating the exact entropy expression, and then integrating, gives

Indeed its derivative is -(1-u)|p|^2/s+p\cdot\E_\sigma b_s(Y-up)/s.

Write r(Y)=D\log(d\sigma/d\mu_s)(Y) in the weak square-root sense. Its mean is

To justify this for finite Fisher information, the Lebesgue density q of \sigma has weak derivative Dq=q(r-DV_s)\in L^1: r is square integrable and DV_s has linear growth. Integrating against expanding cutoffs gives \int Dq=0 and hence Equation (3.19).

In the moving coordinates Y, the score of \sigma_1 relative to \mu_s is

Since 0\le Db_s\le\Id, integration along the segment gives

The mean estimates at the two endpoints therefore imply

Expanding the score square and using Equation (3.19),

The remaining cross term is bounded by C_J\sqrt{aD_0}\,(\sqrt{F_0}+\sqrt{D_0}), and the d-square term by C_JaD_0. Consequently the translated information is finite and satisfies

These calculations require only the weak derivatives already justified, so they apply also when the density of \sigma vanishes. Apply logarithmic Sobolev to \sigma_1, namely F_1\ge2\lambda D_1, and combine Equations (3.18) and (3.21). Since 2/3\le\lambda\le1 on the time interval in question, this proves Equation (3.16).□

Proof. In the first alternative the identities and the last inequality are Lemma 3.5. Apply Lemma 3.6 to \sigma=\eta, where F_0=I and D_0=I/(2\lambda), so the deficit vanishes. The coefficient s^{-2}-\lambda/s is bounded below by a positive constant on J. Thus, with p=\E_\eta Y,

Equation (3.19) applied to \eta now gives |\E_\eta w|^2\le C_J\sqrt a\,I. Since K_s^2\le s^{-1}K_s, the curvature-mean inequality gives the asserted estimate for K_sw.

In the second alternative, compact form embedding supplies a minimizer of the Rayleigh quotient among mean-zero functions of unit norm. Its positive eigenvalue is 1/R_s^P=\lambda. The nullspace consists of constants, since zero weak gradient on \R^n implies constancy. Compactness therefore makes this a positive gap, and interior elliptic regularity makes its eigenfunction \varphi smooth. Commuting a derivative gives A_{\mu_s}D\varphi=(\lambda\Id-K_s)D\varphi. Its right side is in L^2, so Lemma 2.2 gives the operator-domain assertion. Taking the energy pairing and then Poincaré gives, exactly as before,

Coordinates belong to the operator domain because both Y and A_{\mu_s}Y=(Y-b_s)/s belong to L^2(\mu_s). Integration by parts, or self-adjointness with these coordinate tests, yields

Conditional Jensen gives

For every unit vector v, Cauchy–Schwarz thus bounds |\E[\varphi\,v\cdot b_s]| by Ca\|\varphi\|_2. Taking the supremum over v introduces no dimension factor. Since 1-\lambda s\ge1/2 and I=\lambda\|\varphi\|_2^2, it follows that |\E w|\le C_Ja\sqrt I. The bound for K_sw follows from K_s^2\le s^{-1}K_s once again.

It remains to justify the coordinate assertion for the first alternative. Finite H(\eta\mid\mu_s) and the Gaussian tails of \mu_s imply \E_\eta|Y|^2<\infty, by the entropy argument in the proof of Lemma 3.6. The distributional identity

has an L^2(\eta) right side, because b_s is bounded for the fixed law and w\in L^2(\eta). The maximal-domain assertion of Lemma 2.2 applies.□

Proof. Fix s, and let \Phi_{s,t} be the heat continuity flow from Lemma 3.2, for t on either side of s. Transport a form-domain test g by g_t=g\circ\Phi_{s,t}^{-1}. Its distribution under \mu_t is fixed, so both its entropy and its variance are constant. The Jacobian equation \partial_tD\Phi_{s,t}=K_tD\Phi_{s,t}/2 gives

For compact smooth g this follows by differentiating the inverse-transpose Jacobian in the gradient. For a form-domain g, the change-of-variables formula still holds by Sobolev approximation. The flow derivatives and their time derivatives are uniformly bounded on a compact time interval, so the derivative of the integrand is dominated by C|Dg|^2. Dominated convergence proves Equation (3.27) without a smoothness or compact-support restriction on g.

In the strict alternative take g=f and set Q(t)=\Ent_{\mu_t}(f_t^2)/(2\int|Df_t|^2d\mu_t). The denominator is positive near s, and Q(t)\le R_t for both signs of t-s, with Q(s)=R_s. At a differentiability point of R_s, this two-sided contact implies Q'(s)=R_s'. Equation (3.27) and w=2Df/f give Equation (3.26). In the gap alternative take g=\varphi and instead use Q(t)=\Var_{\mu_t}(\varphi_t)/\int|D\varphi_t|^2d\mu_t. Now Q(t)\le R_t^P\le R_t and Q(s)=R_s^P=R_s. The same contact argument gives the identical derivative formula, with w=D\varphi.

Equation (3.23) bounds this derivative by C_J\sqrt a in either alternative, with constants independent of which alternative holds at each time. Lipschitz continuity makes R_s absolutely continuous. For fixed 0<\delta<s<1/2, integration on [\delta,s] gives

First take the counterexample-sequence limit and then let \delta\downarrow0. This proves Equation (3.25).□

Proof. Choose an infinite subsequence with the same alternative at s=1/8. Lemma 3.7 gives all structural identities and the curvature and mean bounds. Proposition 3.8 gives \lambda\to1. Consequently

The energy pairing yields \E|Dw|^2=\lambda I-\E\langle w,K_sw\rangle=(1+o(1))I. Poincaré with constant \lambda^{-1} is the asserted spectral gap. The coordinate and branch-specific identities have already been proved. In the gap alternative all estimates are homogeneous in \varphi, so multiplying it by positive scalars can enforce I\to0 without changing any normalized assertion.□

Proof. Run the stationary diffusion with law \eta, using Brownian noise independent of G', and write P_u=e^{-uA}. Proposition 3.9 puts each coordinate in \Dom(A) and gives the identity

In particular, \E_\eta|Y_0|^2<\infty. The logarithmic Sobolev inequality for \eta then gives

Integrating this inequality over a Gaussian v proves \E_\eta e^{c'|Y_0|^2}<\infty for a sufficiently small c'>0. Here the choice of c' may depend on the individual law and dimension. This exponential-square moment will also permit application of the entropy-loss comparison below.

For a fixed T\le M, set

The Gaussian observation identity gives J_0=TI/2-d(T). Entropy dissipation, including its conditional version, gives -dJ_u\ge F_u\,du, and in particular \int_0^1F_u\,du\le J_0. If \sigma_{u,l}=\Law(Y_u\mid L_T=l), then the entropy chain rule and the strict-case normalization give

For the second equality, write the conditional relative score with respect to \mu_s as the sum of w and the conditional relative score with respect to \eta. The latter scores have conditional average zero at each Y_u, since the conditional likelihoods average to one. This cancels the cross term. At the almost every time where F_u<\infty, the assertion also follows weakly by differentiating that marginal identity: Cauchy–Schwarz gives locally integrable averaged derivatives.

Apply Lemma 3.6 to \sigma_{u,l} and average over l. Since s^{-2}-\lambda/s stays positive, Equations (4.4) give, on [0,1],

where c_s>0 is fixed and \mathcal E is a nonnegative error measure. Indeed the error in that lemma is bounded by a number tending to zero times F_u+I+J_u+I/(2\lambda), whose integral is O_M(I). The measure formulation avoids imposing differentiability on J_u.

The spectral theorem and Equation (4.1) imply

Reversibility gives \E[L_T\mid Y_u]=\sqrt T P_uw(Y_u). Relative entropy with respect to a standard Gaussian is at least half the square of the mean, by testing its variational formula with linear functions. Consequently

Integrating Equation (4.5) with factor e^{2\lambda u} and comparing at u=1 yields

Here the difference between e^{-2\lambda} and e^{-2} contributes only o_M(I).

It remains to turn control of the predicted position into control of the predicted score. The pushforward of \mu_s by b_s has the centered linear moment bound

by conditional Jensen and the corresponding bound for the original signal X. The entropy variational formula, optimized over v, thus gives

Condition Equation (4.3) on L_T and use variation of constants. By Equation (4.6) and the contraction property of conditional expectation,

uniformly for 0\le u\le1. The two scalar functions multiplying the conditional means are linearly independent on [0,1]: the first is nonzero at zero, whereas the second vanishes there and is nonzero for u>0. Their 2\times2 Gram matrix is therefore positive definite. Applying its smallest-eigenvalue bound to each coordinate and then averaging, Equations (4.7) and (4.8) show that

By Lemma 2.4, the left side is 2d'(T) for almost every T. The error is uniform for T\in[0,M], and d(0)=0. Integrating this differential inequality proves Equation (4.2). No lower bound on I is used.□

Proof. The proof adds weak observations that increase the curvature of the base posterior while losing asymptotically no relative entropy. At each step we remove the direction in which the two posterior means differ. The retained output means then agree, and we will show that its relative entropy, averaged over the retained history, is O(\delta^2) for observation strength \delta. Independent copies make the cost of removing these directions vanish after division by the number of copies. That last comparison concerns the score residuals after full observations, whose conditional independence will be essential.

We keep the one-copy problem fixed. The three successive limits will be refinement of a finite observation mesh, increase of the number of independent copies, and increase of the total additional precision. Set b=1/z and take N independent copies. Write Y=(Y_1,\ldots,Y_N) and W=(w(Y_1),\ldots,w(Y_N)); the two possible priors are p^{\otimes N} and q^{\otimes N}. Expectations in the proof use p^{\otimes N}.

Additional observations and their entropy cost. First observe Z_0=\sqrt bY+G_0. The chain rule says that the remaining posterior relative entropy, averaged over this observation, is N(H_0-H_z). Fix u>0 and an integer m, and put \delta=u/m. For i=0,\ldots,m-1, let \mathcal H_i be the retained history before step i. Compute the difference of the two posterior means given this history and let P_i be the orthogonal projection onto its span; set P_i=0 when the difference vanishes. Retain only

where all Gaussian noises are independent. The same measurable rule is used under both priors. One may choose versions of the posterior means by their integral formulas; the positive densities make the rule well defined on all histories of positive probability.

For each realized retained history the likelihood, as a function of Y, is a Gaussian quadratic factor of precision

There is no extra likelihood factor from the choices of P_i: each choice was already determined by the retained past. The factor is the same for both priors. Thus the posterior base has curvature at least B, and the posterior relative score is still W.

We next bound the relative entropy lost in the additional observations. Condition on a retained history at one step and put Q=1-P_i. The projected posterior means agree. After subtracting their common mean, let X_c denote either centered projected signal in \operatorname{ran}Q, whose dimension is at most Nd. The base posterior has curvature at least b\Id. Its centered linear moment bound and integration over a Gaussian vector give

For the alternative, write a_i=\E_p[Y\mid\mathcal H_i]. For any sufficiently small \kappa>0, conditional Jensen gives

Squaring and applying conditional Jensen once more shows that its square, averaged over \mathcal H_i, is bounded by \E_p e^{8\kappa|Y|^2}<\infty. These bounds hold for every history sigma-field, so their constants do not depend on the mesh or on the adaptive choices.

Let h_p,h_q be the output densities of \sqrt\delta X_c+G in \operatorname{ran}Q and let \phi be standard Gaussian density there. Centering and Jensen give the pointwise bound

For either conditional signal, using a conditionally independent copy X_c' and integrating the Gaussian kernel gives

The linear term vanishes because X_c is centered. The bound |e^t-1-t|\le t^2e^{|t|}/2, followed by |X_c\cdot X_c'|\le(|X_c|^2+|X_c'|^2)/2, bounds the last expression by C\delta^2 times the square of a small conditional exponential-square moment. The preceding moment bounds therefore give

This statement includes the zero-dimensional observed subspace, for which the loss is zero. Summing the m steps loses O_{N,b,p}(u\delta) entropy. Apply the anisotropic curvature inequality, Lemma 2.1, to the final posterior pair. For fixed N and u we obtain

Comparison with full observations. We now estimate the effect of the rank-one exclusions. Work in the Hilbert space of square-integrable vector fields on the signal and all the full observations Z_0,\ldots,Z_m. In particular, \|W\|_2^2=NI. Completing the square pointwise gives

Its minimizer is v_*=bB^{-1}W, so \|v_*\|_2\le\|W\|_2. Let \mathcal F_i=\sigma(Z_0,\ldots,Z_i) and let e_i be conditional expectation onto \mathcal F_i. The sigma-fields are nested, and P_i is \mathcal F_i-measurable because the retained history is a function of this full history. We compare the original minimum with one that penalizes the part of v left unpredicted by the full observations: replace 1-P_i by 1-e_i, and denote the resulting minimum by M_0. The nested projections make this new minimum explicitly computable in terms of Gaussian prediction. We first find its minimizer, then use its residuals in a dual test to bound the original minimum below in terms of M_0. Its minimizer is

Here is its explicit form. Set \alpha_k=b/(b+k\delta) for 0\le k\le m. On \operatorname{ran}e_0 the multiplier is one; on \operatorname{ran}(e_k-e_{k-1}) it is \alpha_k; on \operatorname{ran}(1-e_{m-1}) it is \alpha_m. Consequently

All coefficients are nonnegative and sum to one. Put g_i=(1-e_i)v_0. Equation (4.13) gives

The weights in this formula sum to \alpha_{i+1}\le1.

We need a bound uniform over every subdivision. The retained histories need not factor across copies. We use independence only for the full observation arrays: those belonging to different copies are independent under p^{\otimes N}. Conditioning on \mathcal F_i conditions each such array on its own past. Thus e_kW_j depends only on copy j’s observations through step k. Conditional on \mathcal F_i, the blocks g_{i,j} are independent and centered, even though they involve future full observations. To see this directly, factor the joint law into the independent copy arrays and condition each factor on its own past; the resulting conditional kernel remains a product. By conditional Jensen applied to Equation (4.14),

For example, every term with k>i is bounded in conditional squared mean by \E[|W_j-e_iW_j|^2\mid\mathcal F_i] by the tower property; the nonnegative weights of total at most one preserve this bound.

The right sides in Equation (4.15) are uniformly integrable over all these sigma-fields. Explicitly, if Z\ge0 is integrable and M=\E[Z\mid\mathcal F], then

Indeed M\le K/2+\E[Z\1_{\{Z>K/2\}}\mid\mathcal F], and on \{M>K\} the last conditional expectation is at least M/2. The conditional covariance of g_i is block diagonal. Since P_i is measurable in \mathcal F_i and has rank at most one,

The maximum is at most K+\sum_jZ_{i,j}\1_{\{Z_{i,j}>K\}}. Equations (4.15) and (4.16) show that, uniformly in i,m, and \delta,

No moment of the score beyond its second moment is required. In particular this bound permits P_i to depend adaptively on all copies.

For completeness, the comparison of the two minima follows from a dual test. For every v,

The normal equation for v_0 is b(W-v_0)=\delta\sum_i g_i. Therefore the minimum over v of

is attained at v_0 and equals M_0. Use the displayed linear bound at v_* in Equation (4.12). By Equation (4.17), the true minimum is at least

The prediction integral. It remains to compute M_0 and take the limits. The same orthogonal decomposition into nested projection increments gives

Indeed a component first seen after j increments has penalty bj\delta/(b+j\delta), and

Full observations through step i have the sufficient Gaussian statistic of precision b+i\delta. Completing the likelihood square proves sufficiency directly. Hence

This squared prediction norm is nondecreasing with precision, since a weaker observation can be obtained by adding independent Gaussian noise. It is bounded by I, so Equation (4.19) converges as m\to\infty to

First take this mesh limit at fixed N,u. It removes the entropy error in Equation (4.11), while Equation (4.18) remains uniform in the mesh. Divide by N, then let N\to\infty, using q_N\to0. Equations (4.11) and (4.12) give

Finally let u\to\infty. Since 1/b=z and the integrand is bounded by I, this proves Equation (4.9) in the claimed order of limits.□

Proof. For 0\le r<1/4 put

Proposition 3.8 gives \theta(r)=e^{-r}+o(1), uniformly on compact intervals under consideration. The common heat flow dissipates relative entropy at one half the relative Fisher information. Thus its entropy relative to \mu_{s+r} is at most \theta(r) times its initial entropy, by the logarithmic Sobolev inequality at the contemporaneous base law. This assertion applies to conditional laws after averaging as well.

In the strict case write H_r=H(\Law(U_r)\mid\mu_{s+r}) and H_0=I/(2\lambda). The entropy of U_r given L_T, relative to \mu_{s+r} and averaged over the label, is at most \theta(r)(H_0+TI/2), since initially it is H_0+\MI(Y_0;L_T). The chain rule gives

The first equality is Lemma 2.4, conditionally on U_r. Fix v\ge0 and then T>v. Since p_t(r) is nondecreasing in signal strength t, Lemma 4.1 yields

The sequence limit is taken with T fixed. Allowing this fixed T to be arbitrarily large proves

Lemma 4.2 applies to p=\eta and q=\mu_s; the exponential-square moment was proved in the proof of Lemma 4.1. Since H_z\le\theta(z)H_0, it gives

The reverse limiting inequality follows from Equation (4.21), 0\le m\le1, and reverse Fatou. Hence

The function m is nonincreasing in r. For 0<h<r with r+h<1/4, its value at r is bounded below and above by its averages on [r,r+h] and [r-h,r], respectively. Equation (4.22) determines the limits of both averages; letting h\downarrow0 gives Equation (4.20). Finally m(r)\le p_T(r) and Equation (4.21) give the assertion for every fixed T.

We now prove the gap assertion without using additional probes. First linearize Lemma 4.2. Let q be a bounded smooth centered test under \mu_s, with bounded gradient, and let

This alternative has a finite exponential-square moment. Write \bar q_r=\E_{\mu_s}[q(Y_0)\mid U_r] and \overline{Dq}_r=\E_{\mu_s}[Dq(Y_0)\mid U_r]. The output density ratio is 1+\varepsilon\bar q_r, while the predicted initial relative score is exactly

Consequently its squared norm averaged under the alternative output is \varepsilon^2\E_{\mu_{s+r}}[|\overline{Dq}_r|^2/ (1+\varepsilon\bar q_r)]. The entropy expansion H((1+\varepsilon q)\mu_s\mid\mu_s) =\varepsilon^2\|q\|_2^2/2+o(\varepsilon^2), and its output version, now give

Indeed the denominator is bounded away from zero uniformly in r, and conditional Jensen bounds the integrand by a constant times \|Dq\|_2^2, justifying the limit. Approximation in the form norm extends Equation (4.23) to q=\varphi: centered compact smooth approximants have the required boundedness, and all conditional expectation maps are L^2 contractions. To retain compact support while centering, subtract the approximant’s mean times one fixed compact smooth function of mean one; this correction tends to zero in the form norm.

For clarity, we also give the heat variance identity used here. For a centered scalar or finite-dimensional vector field g\in L^2(\mu_s), let g_r=\E[g(Y_0)\mid U_r] under the base law. Then

For bounded g, both the numerator density g_r\mu_{s+r} and the denominator density \mu_{s+r} solve the heat equation, and the quotient calculation with the square function gives (\partial_r-\Delta/2)(\mu_{s+r}|g_r|^2) =-\mu_{s+r}|Dg_r|^2. Integrate against expanding cutoffs, whose Laplacians tend uniformly to zero. The bound |g_r|\le\|g\|_\infty removes the cutoff terms and gives the integrated identity on positive time intervals. Its initial endpoint follows from \|g_r(U_r)-g(Y_0)\|_2\to0: for a bounded continuous test use g(U_r) as a predictor and dominated convergence under the coupling U_r=Y_0+\sqrt rG, and then approximate in L^2(\mu_s) and use conditional contraction. Approximate a general L^2 field by bounded ones. Applying the same identity to their differences shows that their gradients are Cauchy in the space-time energy norm, while conditional contraction gives convergence of both endpoint norms. This proves the integrated version of Equation (4.24) in full generality.

Poincaré at \mu_{s+r} and preservation of the mean therefore imply

Apply this first to \varphi, and then to w-\E w. The latter application and Equation (4.1) give m(r)\le\theta(r)+o(1). The former and \|\varphi\|_2^2=I/\lambda, inserted into Equation (4.23), give \int_0^z m(r)\,dr\ge(1-\theta(z))/\lambda. The same integral squeeze and monotonicity argument prove Equation (4.20). Multiplying \varphi by any positive scalar multiplies every squared quantity by its square, so we may finally choose these scalars to arrange I\to0.□

Proof. We first prove the strict-case spectral estimates. If a random vector S has covariance C and mean a, the affine estimator

of S from S+G is its least-squares affine predictor. Its mean squared norm is |a|^2+\tr(C^2(\Id+C)^{-1}), by direct expansion of the covariances. Conditional expectation has at least this squared norm, because it is the orthogonal projection onto all functions of the observation. The same calculation holds conditionally on any side information, using fresh independent Gaussian noise.

By the Gaussian signal derivative formula and its monotonicity,

Apply the affine prediction bound to S=w and C=\Cov_\eta w. For every fixed \epsilon>0,

since t^2/(1+t)\ge\epsilon t/(1+\epsilon) for t>\epsilon. For every projection of rank at most cI,

First take the sequence limit, then \epsilon\downarrow0. Equation (4.1) proves Equation (4.25) uniformly in \Pi. In the gap case a projection of rank at most cI is eventually zero, which proves the same assertion.

Conditionally on U_r, the additional observation L_1 has signal w and covariance C_r. Orthogonality of successive conditional expectations and the conditional affine prediction bound yield

Proposition 4.3 makes the left side o(I). The same scalar eigenvalue inequality proves Equation (4.27).

We finally bound the dimensions of the deterministic outlier spaces. Let

where the inner posterior kernel always refers to the original prior \mu. Conditional Jensen and the definition of M_s imply

For each deterministic projection Q of rank l\ge1, the original subgaussian assumption gives

Indeed choose a 1/2-net of the unit sphere in \operatorname{ran}Q with at most 5^l points. The norm |QX| is at most twice the largest absolute linear form over this net, and bounding the exponential of a maximum by the sum proves the claim.

In the strict case, form the joint law by first drawing Y_0 from \eta and then drawing X from the original posterior kernel. Its entropy relative to the original joint law of (X,X+\sqrt sG) is exactly H_0=I/(2\lambda). Data processing therefore bounds the entropy of its X-marginal relative to \mu by H_0. Applying the entropy variational inequality to Equation (4.30) gives

If the spectral subspace of D above \tau>0 has dimension l, then \tau l\le Ca^2(H_0+l). For sufficiently small a this implies l\le2Ca^2H_0/\tau=O_\tau(I); the assertion is trivial when l=0. The spectral subspace of \E ww^* above \epsilon has dimension at most I/\epsilon, since its trace is I. Take V_\epsilon to be the span of this subspace and the spectral subspace of D above s\epsilon. Its dimension is at most C_\epsilon I, and Equation (4.29) gives all three bounds in Equation (4.26).

In the gap case, averaging the original posterior under \eta=\mu_s gives D=\E_\mu XX^*\preceq Ca^2\Id. Thus Equation (4.29) gives the two claimed operator bounds directly. Also \|\E ww^*\|_{\op}\le I\to0, so V_\epsilon=\{0\} suffices eventually. This completes all assertions.□

Proof. The first equality is conditional expectation:

For fixed y, Gaussian integration by parts identifies the order-j Hermite coefficient of h_r(y+\sqrt r\,G) as r^{j/2}D^jg_r(y). Its orthogonal projection onto that chaos is

where the vector output slot is not contracted. By Equation (5.9), the squared norm of this projection is r^j|D^jg_r(y)|^2/j!. Bessel’s inequality, integrated in y\sim\eta, proves the asserted sum.

These calculations require only \E|h_r(Y+\sqrt rG)|^2<\infty. To justify the derivatives, on a compact set of starting points use positivity of \eta on a slightly larger compact set. Integrating translated Gaussian kernels over that larger set bounds each fixed differentiated kernel by a constant times the mixture density of U_r. The enlargement absorbs the polynomial factors from differentiation. Thus the kernel integrals are classically smooth locally; alternatively all identities first hold distributionally. The Bessel bounds for orders j and j+1 put D^jg_r and its weak derivative in L^2(\eta), hence in the form domain by cutoff and local smoothing. The local constants need not be uniform in the dimension: only the exact integrated Bessel inequality is used quantitatively.□

Proof. The map u\mapsto u\cdot Z from Euclidean space to scalar L^2 has operator norm at most \sqrt c. Apply its adjoint to each scalar coefficient of F and sum the squared bounds. This proves Equation (5.11) without a factor equal to the number of coordinates of either variable.

For the second assertion, sum the coefficient spectral measures of S-\E S and call the resulting measure \nu. Then

where the left side is nonnegative. For every fixed \delta>0, the spectral mass and the spectral energy outside [1-\delta,1+\delta] are o(I): the part below 1-\delta is absent eventually, and on x\ge1+\delta both 1 and x are bounded by a \delta-dependent constant times x-\lambda. The constant component has squared norm o(I) as well.

On the remaining interval the multipliers e^{1-x}-1 and (x-1)e^{1-x} tend uniformly to zero as \delta\downarrow0. Outside it, (1+x)|e^{1-x}-1|^2\le C(1+x) and (x-1)^2e^{2-2x}\le C. Integrate these inequalities against the spectral measure, first take the sequence limit and then let \delta\downarrow0. This gives Equation (5.12). The same functional calculus shows that e^{1-A}S belongs to every \Dom A^p.□

Proof. Start with T_0=w. Suppose the assertions have been constructed through level l, and put L=l+1 and V=DT_l. Self-adjointness gives

If \Pi is an admissible deterministic projection acting on an existing tensor slot, it commutes with A and differentiation. Consequently

The error is uniform over projections of rank at most cI for fixed c.

Fix 0<r<1/4. Set V_j=T_j for 0\le j\le l and V_L=V. Successive form integration by parts gives

Indeed the assertion at j=0 is Equation (5.10). For j\le l,

Replacing DT_j by T_{j+1} when j<l costs o(I). All test derivatives and their form energies are finite by Lemma 5.2; its bound controls every error for fixed r and j. No uniform estimate as r\downarrow0 has been used.

Here is the precise Bessel budget. Equation (5.14) and \|V_j\|_2^2/I\to1 imply

Multiply Equation (5.10) by e^{2r}/I and subtract these minimum contributions. The resulting bounds are

For the first line, expand the squared difference and use the correlation at order L; its normalized limit subtracts exactly one from the normalized squared derivative norm. Thus the exponential-series tail accounts for the entire excess, rather than merely bounding the norms.

Choose r\downarrow0 sufficiently slowly along the sequence in these two estimates. The field S_0=e^rD^Lg_r then satisfies

It is symmetric in its last L slots; V is symmetric in its first L slots. For L\ge2, the permutations of those two sets of slots generate every permutation of the L+1 slots. Indeed their adjacent transpositions together are all the adjacent transpositions. If a permutation fixes S_0, its action changes V by at most 2\|S_0-V\|_2; a permutation fixing V costs zero. Expressing each permutation as a fixed finite word therefore shows that V is o(\sqrt I) from its full symmetrization. For L=1, V=Dw is already symmetric because w is a gradient. Let S be the full symmetrization of S_0. Averaging coefficient permutations is an orthogonal contraction and commutes with differentiation, so

The preceding projection estimate for V now holds on its last slot too, since a coefficient-slot permutation moves it to an old slot with an o(\sqrt I) error uniform over projections.

To apply Poincaré sharply we still need to center V. Write C_y=\Cov(T_l,y), C_b=\Cov(T_l,b), and C_\omega=\Cov(T_l,\omega), with the vector variable occupying the last slot. Testing the equation for AT_l against centered coordinates gives

For clarity, the first exact identity before the error is \E V=\E[(AT_l)\otimes(y-\E y)]. Poincaré gives \Cov_\eta y\le\lambda^{-1}\Id, and Equation (5.11) bounds its error by \lambda^{-1/2}\|(A-1)T_l\|_2. This is the needed dimension-free estimate. The second identity follows by integration by parts using Ay=(y-b)/s-\omega. These coordinate tests are legitimate in the form domain; their distributional A-images are in L^2 by the stated second-moment assumptions, hence they are also in \Dom A.

Project the last slot onto 1-\Pi_\epsilon. The two covariance terms involving b and \omega have norm at most C\sqrt{\epsilon I} by Equations (5.3) and (5.11). Subtracting the two identities in Equation (5.18), and using s\ne1, therefore bounds this part of \E V by C_s\sqrt{\epsilon I}+o(\sqrt I). On the \Pi_\epsilon part,

by the projected-energy estimate and \rank\Pi_\epsilon\le C_\epsilon I. First take the sequence limit and then let \epsilon\downarrow0. This proves |\E V|=o(\sqrt I), and thus |\E S|=o(\sqrt I).

Lemma 5.3 applies to S. Define

It is symmetric, lies in every operator-power domain, is o(\sqrt I) from V, and satisfies the strong near-eigenfunction estimate. Its mean is e\E S=o(\sqrt I). Finally, coefficient projections commute with the semigroup, and \|e^{1-A}\|_{2\to2}\le e. Together with S-V=o_{L^2}(\sqrt I), this preserves the uniform projected-energy estimates. The induction is complete. Each step used only a finite collection of limits; diagonal choices give the assertion for all fixed levels.□

Proof. The operator-domain Itô formula from Lemma 2.2 gives, for each fixed d,

Stationarity, Cauchy–Schwarz for the time integral, and Itô isometry show that the last two terms are o(\sqrt I), uniformly for v\in[0,T] in squared-mean norm. Substitute this formula for the integrand at the next level, and repeat m+1 times. For fixed l,m,T, every replacement error remains o(\sqrt I); iterated isometry sums coefficient squares and introduces no dimension factor.

After multiplication by e^{-t}, the remaining integral has coefficient 2^{(m+1)/2}, is over the ordered time simplex of length t, and has innermost integrand e^{u-t}T_{l+m+1}(Y_u). Since u\le t, its squared norm is at most

This proves the remainder estimate and its uniform version. The other terms have coefficients evaluated at Y_0. Symmetry in their Brownian slots turns their ordered iterated integrals into t^{j/2}\mathfrak h_j(N)/j!. To verify the factor directly, expand the Brownian exponential martingale \exp(\xi\cdot B_t-t|\xi|^2/2) by iterated Itô integration and compare its coefficients with Equation (5.8). The additional \sqrt2 at each stochastic integration gives exactly (2t)^{j/2}/j! in Equation (5.19).□

Proof. We first prove the assertion for Q. Couple the heat observations by U_v=Y_0+W_v, where W is a standard Brownian motion independent of Y_0\sim\eta, and retain h_v=\E[w(Y_0)\mid U_v]. Put H_v=Dh_v(U_v) wherever this derivative is square integrable. The conditional covariance has the exact representation

To check the first identity and its normalization, let p_v denote the density of U_v. The entries of p_vh_v solve the heat equation with generator \Delta/2. Direct differentiation gives the matrix identity

Integrate it against the later heat kernel and divide by p_r. The initial term becomes \E[ww^*\mid U_r], and the final term is h_rh_r^*, which proves the first formula. Taking the trace and expectation proves the second formula. For nonsmooth or unbounded initial w, perform the calculation on scalar projections with convex truncations increasing to the square, and then use polarization. Spatial cutoffs and the square-integrability of w justify the positive-time passage. At time zero, approximation of w in L^2(\eta) by bounded continuous functions gives \E|w-\E[w\mid U_v]|^2\to0. Thus there is no missing boundary term in Equation (5.22).

By Equation (5.1), \|Q\|_2^2=\langle w,Aw\rangle_2=(1+o(1))I. For almost every v>0, convolution differentiation and form integration by parts give

The error is uniform in v: Jensen’s inequality gives \|g_v\|_2\le\sqrt I, so it is bounded by \|(A-1)w\|_2\sqrt I. The convolution derivative identity follows first with cutoffs and bounded functions, then by the square-integrable derivative bounds in Equation (5.22).

Fix a small z>0 with 2z<1/4. Combining the last two equations and using 0\le m(v)\le1 yields the exact limiting discrepancy

Dominated convergence is applicable to m(v), so this calculation does not require uniform convergence of the prediction curve near zero.

We select a time in [z,2z] while preserving a covariance-tail bound. For a fixed \kappa>0, define

For fixed r\in[z,2z], Equation (5.4) gives a(r)\to0, and a(r)\le1/z follows from \E\tr C_r\le I. Consequently \int_z^{2z}a(r)\,dr\to0. On the other hand, Equation (5.25) and Markov’s inequality show that the set of times satisfying

has measure at least z/2, with an absolute C and for sufficiently far terms of the sequence. The set where a(r) exceeds a quantity tending to zero has measure tending to zero. Select r\in[z,2z] in the intersection. Thus r may vary with the sequence index, but

All constants in this selection are for fixed z,\kappa.

We compare B=H_rH_r^* with C=C_r/r in trace norm. For matrices U,V,

Compare both B and C with \E[QQ^*\mid U_r]. For B, conditional Jensen’s inequality and the first bound in Equation (5.26) give an O(\sqrt z)I bound on the expected trace-norm difference. For C, use the first identity in Equation (5.22), followed by Equation (5.27) and Cauchy–Schwarz in both probability and time. The integrated discrepancy is O(z^2)I, whereas

Here we used r\le2z and the nonnegative integral in the second identity of Equation (5.22). The normalized limsup of the right side is O(z), by prediction at the fixed time 2z. No prediction limit at the selected varying time is needed. Division by r\ge z therefore also costs at most O(\sqrt z)I. We have proved, in normalized limsup,

Fix \delta>0, take \kappa=\delta^2/2 above, and set P=\1_{\{B>\delta^2\}}. Positivity of C implies

Since \delta^2\rank P\le\tr(BP), the first term is absorbed into the left side. Equations (5.26) and (5.28) bound the remaining high singular-value energy by O(\sqrt z)I+o(I). Clip H_r at \delta; its squared distance from the operator-norm ball of radius \delta is at most this energy. The first estimate in Equation (5.26) then gives the same bound, up to constants, for the distance of Q from that ball. Let z\downarrow0 after the sequence limit. The squared distance is o(I) for every fixed \delta>0. Applying this distance assertion at cutoff \delta/2 bounds the energy of singular values above \delta by four times the squared distance, proving Equation (5.21) for Q. Since \|T_1-Q\|_2=o(\sqrt I), it also proves the assertion for T_1.

It remains to propagate this estimate to higher levels. Fix l\ge1 and t>0. Apply Lemma 5.5 with starting level 1. Conditional on Y_0, the Hermite chaoses are orthogonal even after applying an arbitrary measurable projection P(Y_0) to the first tensor slot. The chaos containing T_l(Y_0) has squared coefficient

Use the expansion to order m\ge l-1, first take the sequence limit, and then let m\to\infty in its remainder bound. The resulting inequality is

The remainder estimates are uniform over such projections, since their operator norms are at most one.

Choose P(Y_0) to project onto the singular values greater than a fixed \epsilon>0 of the first-slot flattening of T_l(Y_0). Then

For every \delta>0, let S_\delta be the singular-value clipping of T_1 at \delta. Pointwise,

Stationarity controls the o(I) clipping error for T_1(Y_t). Hence the normalized limsup of the first expectation in Equation (5.29) is at most \delta^2/\epsilon^2. Send \delta\downarrow0 to obtain the tail claim at threshold \epsilon for T_l. No independence between P(Y_0) and T_1(Y_t) was required. Tensor symmetry supplies every other one-slot flattening.□

Proof. For compactly supported smooth P, commuting two first derivatives in weighted integration by parts gives

Cauchy–Schwarz applied to the two arrays with i,j interchanged bounds the sum by \|DP\|_2^2, without a dimension factor.

For the stated P, membership in \Dom A gives P\in H^2_{\rm loc} by local elliptic regularity and DP\in L^2(\eta). Choose smooth cutoffs 0\le\chi_R\le1 tending to one with \|D\chi_R\|_\infty\to0. Equation (5.32) applies to \chi_RP by local Sobolev approximation. Since

its right side is bounded. The second convergence uses |\langle P,HP\rangle|\in L^1, a consequence of P,HP\in L^2. Weak compactness in L^2 and distributional convergence of D^*(\chi_RP) identify the limit as D^*P. Lower semicontinuity proves Equation (5.30). Only first derivatives of the cutoffs have entered the estimate.

For compactly supported smooth F, differentiate the scalar operator:

Pair this identity with P, use symmetry of H, and integrate by parts to obtain Equation (5.31). For general F\in\Dom A, use the operator core from Lemma 2.2 to choose compact smooth F_j with F_j\to F and AF_j\to AF in L^2. This convergence also holds in the form norm, since

All terms in Equation (5.31) therefore converge: the three fixed factors are D^*P, AP, and HP, each in L^2. This proves the claimed global identity.□

Completion of the proof of Proposition 5.1. Lemma 5.4 supplies the fields and all assertions except Equation (5.6). Fix l\ge0 and write F=T_l. View DF as a matrix with its derivative slot as row index and all old tensor slots as column index. We construct a matrix field P satisfying

where Q=Dw in the strict branch and Q=0 in the gap branch.

In the strict branch, apply Lemma 5.6 to the corresponding flattening of T_{l+1}. Choose a singular-value cutoff \delta\downarrow0 sufficiently slowly that the clipped field C obeys

Set P=e^{1-A}C=eP_1C. Markov averaging and convexity of the operator-norm ball give \|P(y)\|_{\op}\le e\delta. The spectral multipliers e^{1-x}, (x-1)e^{1-x}, and \sqrt x\,e^{1-x} are bounded for x\ge0. Moreover |e^{1-x}-1|\le C|x-1| there. These bounds, the approximation of C, and the near-eigenfunction assertion for T_{l+1} give the first two estimates in Equation (5.33). Finally \|Q\|_2^2=(1+o(1))I and

The clipping is only used to construct P, so it need not preserve tensor symmetry. In the gap branch take P=T_{l+1} directly. The first two estimates follow from Equation (5.5), and the last is immediate from Q=0. No operator-tail argument is needed in this branch.

The drift hypothesis gives H=D^2(-\log\eta)=K-Q in either case. Both KP and QP are in L^2, and the operator-domain assertion for P was retained by its construction. Lemma 5.7 therefore applies. In particular,

Insert AF=F+(A-1)F and AP=P+(A-1)P into Equation (5.31). The terms \langle F,D^*P\rangle_2 and \langle DF,P\rangle_2 cancel. Consequently

Since \|K\|_{\op,\infty}\le s^{-1} and P-DF=o_{L^2}(\sqrt I), this gives

Replacing DF by T_{l+1} costs o(I) by the derivative estimate. It proves Equation (5.6) at level l+1 on the derivative slot, and hence on every slot by symmetry. Level zero was an initial assumption. All claims of the proposition follow.□

Proof. For rectangular matrices X,Y, the Araki–Yamagami inequality [1, Theorem 1] gives the following square-root bound. We give a finite-dimensional proof by self-adjoint dilation:

Indeed the absolute value of the dilation of X has diagonal blocks \sqrt{XX^*} and \sqrt{X^*X}. If A,B are self-adjoint and f is scalar Lipschitz, expansion in an eigenbasis of A and an eigenbasis of B shows that \|f(A)-f(B)\|_{\HS}\le\operatorname{Lip}(f)\|A-B\|_{\HS}. Apply this with f(x)=|x|. Difference quotients, followed by form approximation, prove that R is in the form domain and that \|DR\|_2^2\le2\|D\Psi\|_2^2.

The curvature conclusion of the hierarchy yields

We also have

To see the latter assertion in the strict branch, fix \delta>0 and let P_\delta be the spectral projection of \E M_s onto (\delta,\infty). Lemma 4.4 gives \rank P_\delta=O_\delta(I); the deterministic-projection conclusion of Proposition 5.1 therefore gives \tr(P_\delta\E R^2)=o(1). Since 0\le M_s\le1, the left side of Equation (6.4) is at most o(1)+\delta\E\tr R^2. Send \delta to zero after the sequence limit. In the gap branch use directly \|\E M_s\|_{\op}=o(1).

Let R' be evaluated at an independent point. Insert 1-M_s+M_s between R and R' in \|\E R\|_{\HS}^2=\E\tr(RR'). Hilbert–Schmidt Cauchy–Schwarz bounds the term containing 1-M_s by

The other term is bounded by

Poincaré for \eta, whose constant tends to one, now gives the lower bound on \|DR\|_2^2 from Equation (6.2).□

Proof. For a component \psi_d with d slots, put

Symmetry and Cauchy–Schwarz give |c_d|\le a_d, and direct expansion gives

It remains to prove |\sum_d c_d|\le\tr(RHRH). This correlation bound can also be deduced from the Wigner–Yanase case of Lieb’s trace-concavity theorem [15]; we give a proof by weighted interpolation.

Define the linear map C by

Flattening all remaining indices exhibits C(Y) as a sum of terms A_\alpha Y^{\mathsf T}A_\alpha^*. Thus C is positive and C(1)=C^*(1)=\rho. Restrict to the support of \rho, on which inverse powers are defined, and set

The map U is unital and contractive in operator norm: its preceding representation is an isometric compression of copies of Y^{\mathsf T}. It also satisfies \tr(\rho U(Y))=\tr(\rho Y). For 0\le\Re z\le1 introduce

On \Re z=0, multiplication on either side by imaginary powers of \rho is isometric, so U_z is operator-norm contractive. At z=1,

Its adjoint is a unital positive map with the same compression and transpose representation, hence is operator-norm contractive. Duality makes U_1 trace-norm contractive; imaginary powers give the same conclusion on \Re z=1.

The interpolation used here is the finite-dimensional weighted Schatten argument; compare [3, Section III, Theorem 4]. We give the details for our positive correlation map. Given matrices of Hilbert–Schmidt norm one, continue the singular values of the input by powers 2z, and those of the matrix paired with its output by powers 2(1-z), keeping the unitary factors fixed. The bilinear trace pairing is analytic in the strip, bounded by one on both boundary lines by the preceding endpoint bounds and trace duality. The three-lines inequality bounds it by one at z=1/2. Zero singular values follow by approximation. This proves \|U_{1/2}(Y)\|_{\HS}\le\|Y\|_{\HS}. Take Y=\rho^{1/4}H\rho^{1/4}. Then

Cauchy–Schwarz proves the claim. Compression to the support leaves the correlation unchanged: every tensor slot lies in that support. Complexification, used only for interpolation, preserves the conclusion for real tensors.□

Proof. In an eigenbasis of R with eigenvalues r_a, define (H_i)_{ab}=2(D_iR)_{ab}/(r_a+r_b) when r_a+r_b>0, and zero otherwise. The kernel–kernel block of D_iR vanishes almost everywhere. One way to check this for a form-domain positive matrix field is to use absolutely continuous line representatives: at a differentiability point, the quadratic form on every vector in the kernel has a local minimum zero, and polarization gives the assertion. The formula is measurable and gives

Set

The Sylvester equation implies the exact identities

Differentiating R^2=\Psi\Psi^* in the weak form sense gives

For example, the first equality follows by pairing D_i(R^2) with H_i/2; all products are integrable by the displayed bound. The second uses the symmetry of each D_i\Psi on its existing slots. Lemma 6.2 and Cauchy–Schwarz yield

Since a\le4e, d\to1, and \liminf e\ge1, this first forces e\to1 and then c\to0; Equation (6.9) gives a\to1. Finally,

Both tend to zero. Symmetry gives the assertion for the other slots.□

Proof. We repeatedly use the dimension-free inequality

By Equation (6.8), the first sum in Equation (6.10) is close in expected trace norm to the one-slot reduction of D\Psi, with the derivative index included among the column indices. The derivative coupling identifies this with

The trace-norm error from the coupling is o(1) by Equation (6.12) and Equation (6.2). Shifting the band changes its reduction only by its two endpoints, whose expected trace-norm cost is

This proves the first balance equation. Comparing the arrays of matrices (H_iR)_i and (RH_i)_i proves the second by the same inequality and the summed commutator estimate.

For the original, possibly unbounded column write \mathbf H v=(H_iv)_i, S=\mathbf H^*\mathbf H, and J=(1+S)^{-1}. Let \mathbf K=\mathbf HR-R^{\oplus n}\mathbf H, so that \E\|\mathbf K\|_{\HS}^2=o(1). Algebra gives

The bounds

therefore imply \|\mathbf H[R,J]\|_{\HS}\le2\|\mathbf K\|_{\HS}. For every fixed f(x)=a+b/(1+x), it follows that

The squared norms of \mathbf Hf(S)R and \mathbf HRf(S) differ by o(1): the latter has bounded mean-square norm, and the preceding bound controls their difference. Testing the second balance equation against the bounded matrix f(S)^2 consequently gives

Use f=1,1+J,1-J and (S-1)^2/(1+S)=(S-1)-2(S-1)J to obtain

All these expressions are integrable: their absolute values are bounded by a constant times \tr(R^2(1+S)).

Let Q=\1_{(4,\infty)}(S) and P=1-Q. On this spectral region, (S-1)^2/(1+S) dominates both a positive constant and a positive multiple of S. Thus

The input truncation costs \sum_i\E\|H_iQR\|_{\HS}^2=\E\tr(R^2SQ)=o(1). The output truncation costs

Consequently replacing H_i by PH_iP has total right-R squared cost o(1). The new column obeys \sum_i(PH_iP)^2\le PSP\le4P.

Here is the perturbation accounting needed both now and below. If symmetric \widetilde H_i have

then symmetry gives the same bound with R on the left. Each derivative error in Equation (6.8) changes in norm by at most \sqrt\Delta, and the summed commutator norm changes by at most 2\sqrt\Delta. The two balance errors change in expected trace norm by at most

by Equation (6.12), and this is o(1). The new summed energy is bounded by the old one plus the perturbation in norm. These statements apply to the compression just made.

For the bounded column, [S,R]=\mathbf H^*\mathbf K-\mathbf K^*\mathbf H satisfies \|[S,R]\|_2\le4\|\mathbf K\|_2=o(1). Put T=S-1, so \|T\|_{\op}\le3. The second balance equation says \E\|RTR\|_1=o(1). Finally,

The expectation of the first term is at most 3\E\|RTR\|_1 in absolute value; the second is at most 3\|[T,R]\|_2\|R\|_2=o(1). This proves the last assertion.□

Proof. Equation (6.8) implies that moving H_i between any two existing slots costs o(1) in summed mean square. Three slots suffice to exchange two factors. In the following chain, each \simeq means that the difference has squared norm o(1) after expectation and summation over i,j:

The equalities use commutation on different slots. Each move costs at most the one-factor error multiplied by the column bound: for any tensor v,

The squared norm of the sum of the four errors is at most four times their summed squared norms. Thus the total commutator cost on \Psi is o(1). Its cost on \Psi equals its cost on R, since both have reduction R^2.

For the second conclusion, write \Psi=RJ_0, where J_0J_0^* is the support projection of R. The matrix-valued array Z_{ia,b}=(H_iR)_{ab}-(H_aR)_{ib} has its column index in that support, so \|ZJ_0\|_{\HS}=\|Z\|_{\HS}. The entries of ZJ_0 express the exchange of the derivative index and the first tensor index in (H_i)_{(1)}\Psi. Equation (6.8) and the approximation of D\Psi by fully symmetric T_{l+1} give the claim. For perturbed matrices, the derivative errors and the slot-moving errors have the bounds established after Equation (6.14); the same four moves use the new uniform column bound. The index-exchange norm itself changes by at most 2\sqrt\Delta.□

Proof. Choose an orthogonal eigenframe O for R at the initial point. Transform all three indices: in that frame the matrices are

This preserves the column bound, all the summed norms, and the index-exchange estimate, since these are Euclidean tensor contractions. Finite-dimensional spectral calculus and an ordered orthonormal basis completion give measurable choices, including at repeated eigenvalues.

Let h=\log(1+\epsilon) and take an independent \theta uniform on [0,h). Group positive eigenvalues in [e^{\theta+mh},e^{\theta+(m+1)h}), m\in\mathbb Z, and round those in each nonempty block down to \beta=e^{\theta+mh}. The zero eigenspace is kept separate. Write

For 0<\delta<h, a pair of positive eigenvalues with |\log(r_a/r_b)|\ge\delta satisfies r_b^2\le C\delta^{-2}(r_a-r_b)^2, when \delta\le1. If their logarithmic distance is smaller than \delta, the probability that a block boundary separates them is at most \delta/h. Consequently the right-R squared cost of deleting entries with row and column in different blocks is bounded by

The final term covers a zero row and positive column directly by the commutator. A zero column has no right-R cost. For example, choose \delta\to0 slowly enough that \kappa/\delta^2\to0, and then h\to0 slowly enough that \delta/h\to0.

Deleting entries whose outer index belongs to another block has vanishing cost as well. On the set where i and b have different block labels,

Sum over the indices. The first term is the second defect of Equation (6.15); the second is a row–column block exclusion of the kind already bounded in Equation (6.19). This includes outer indices in the zero eigenspace. Within each block the resulting matrices have the form P_\beta\widehat H_iP_\beta for i\in\beta, and vanish otherwise. Thus

They are symmetric and their total change has the cost in Equation (6.14).

For every block matrix X one has the precise comparison

In particular \tau(1)\to1. The perturbation bounds following Equation (6.14) prove the derivative, commutator, and balance conclusions for these matrices. The bounded-column argument at the end of Lemma 6.4 again gives \E\|(S-1)R\|_{\HS}^2=o(1), so Equation (6.20) proves the first tracial defect. Lemma 6.5 gives the second. For the third, all three indices now lie in one block, and the index-exchange defect is exactly weighted by r_b^2; replacing this by \beta^2 only decreases it. Matrix symmetry and exchange of i,a together give asymptotic symmetry under every permutation of the three indices.

It remains to bound the map from the outer index to matrix coefficients. On each block let

The map G has operator norm at most two because \|Gv\|_{\HS}^2=\langle v,Sv\rangle\le4|v|^2. Total symmetry just established says

Clip the singular values of F to two, obtaining F'=FB for a real positive contraction B on the input space. Metric projection onto the operator-norm ball in Hilbert–Schmidt norm gives \|F-F'\|_{\HS}\le\|F-G\|_{\HS}; this also follows immediately by singular-value decomposition and the variational characterization of singular values. Thus the block-weighted squared cost is o(1). Writing H'_i=\sum_j B_{ji}H_j shows that every H'_i remains symmetric. For each vector v,

so the column bound is preserved, as is the block support. The weighted cost controls the right-R cost by Equation (6.20). Therefore the derivative and balance errors have exactly the bounds stated after Equation (6.14); index exchange changes by at most twice the perturbation norm, and the four-move argument controls the commutators. Repeating the bounded-column last step and then Equation (6.20) proves all three tracial defects for H'_i. The defining singular-value clipping proves Equation (6.18).□

Proof. We first prove a conditional moment bound on one fixed block, allowing a general column bound S\le C\Id. We use the Gaussian trace-moment integration-by-parts argument of [19, Lemma 6.1 and Section 7]; the weighted near-commutation limit will be proved afterward. Write D=\sum_iN_iH_i. Gaussian integration by parts gives

In an eigenbasis of D with eigenvalues d_a, the absolute value of a summand is bounded by

For p>1, the scalar weighted arithmetic–geometric mean inequality bounds the product of powers by

Since \sum_{i,b}|(H_i)_{ab}|^2=S_{aa}\le C and similarly with a,b exchanged, each summand is at most C\tr|D|^{2p-2}. For p=1 the assertion is \tr S\le C\tr1. Induction proves \E_N\tr D^{2p}\le C^p(2p-1)!!\tr1. After multiplying by the block weights, all fixed moments are uniformly bounded. Also e^{u|x|}\le2\cosh(ux), so the even-moment series gives

This proves Equation (6.22) with C=4.

For clarity, the norm \|X\|_{p,\tau}=\tau(|X|^p)^{1/p} is formed on the direct sum of the blocks and then averaged over the classical random variables. Schatten Hölder followed by scalar Hölder gives

These bounds use the actual weights \beta^2, with no factor depending on a block dimension. The finite-matrix Schatten inequality can itself be obtained by the same singular-value three-lines argument as in Lemma 6.2, interpolating the endpoint bounds with one trace norm and the other norms operator norms, and then applying duality.

For independent N_r,N_s,

All its higher fixed moments are bounded by Hölder and the moment bound just proved. Interpolation between L^2(\tau) and an arbitrarily high fixed L^p(\tau) therefore makes this commutator tend to zero in every fixed finite power norm. Applying Hölder to the factors on either side of a commutator shows that an adjacent transposition in a word of fixed length costs o(1) in L^2(\tau). A finite sequence of transpositions proves the reordering assertion.

Now expand a mixed moment by Gaussian pairings, conditional on the matrices. Pairings must join indices of the same Gaussian vector. For each pair introduce its own independent standard Gaussian vector, shared only between the two factors in that pair. This represents the contracted matrix word exactly as a conditional expectation. Reorder the word until each pair is adjacent; the error is o(1) even in L^2(\tau). Averaging the independent vectors then gives S^{m/2}. For each fixed integer b, boundedness of S gives

Each pairing consequently contributes \tau(1)+o(1)=1+o(1). Odd moments vanish. The number of admissible pairings is precisely the scalar independent-normal moment, proving Equation (6.21).

Finally expand the tensor Hermite polynomial using its defining exponential \exp(\langle v,N\rangle-|v|^2/2). A term consists of free N indices and a number of signed Kronecker-delta pairs. Represent each delta pair by its own independent Gaussian vector, as above, and move its two matrix factors together. Conditional averaging is contractive in L^2(\tau), so the reordering error remains o(1). The resulting factors S can be replaced by one, even between fixed powers of D_1, by Hölder and interpolation applied to the bounded matrix S-1. The resulting polynomial is

There are only finitely many terms for fixed j, proving Equation (6.23).□

Proof. The proof has three steps: replace the added tensor indices in the hierarchy expansion by products of the initial matrices, identify their Wick contractions, and sum the resulting scalar Hermite series. All limits involving the degree of that series are taken after the counterexample-sequence limit, with t fixed.

The derivative-action estimate and the hierarchy coupling give

It also holds when the band is shifted by any fixed integer a\ge0. Indeed the same matrices have been constructed using levels k through 2k-1. Removing or adding at most a levels in the error sum costs O_C(a/k), since the unnormalized squared norm of each one-step error is at most

by the bounded column and the diagonal endpoint control. This also explains why the matrices may already include the block and input clipping modifications: their cost on \Psi is the right-R cost in Equation (6.14).

Iterating Equation (6.26), for each fixed j we may replace the extra-index tensor T_{l+j} by

in the normalized band mean-square norm, with error o(1). More explicitly, if e_a is the root mean-square error in the one-step estimate on the band shifted by a, the root error after j steps is at most

Here multiplication by a column of H_i costs at most \sqrt C in the summed coefficient norm. Each e_a=o(1) at fixed a, so the displayed bound vanishes. Contraction with \mathfrak h_j(N) multiplies squared coefficient norm by at most j!: it first symmetrizes the contracted indices, an orthogonal projection, and Gaussian Hermite orthogonality then gives the factor j!.

Apply Lemma 5.5 to the band of levels in \Psi. For each fixed m, the preceding replacements and Equation (6.23) give

where

and

To justify the conversion of a tracial polynomial error to the norm of its action on \Psi(Y_0), use \|X\Psi(Y_0)\|_{\HS}=\|XR\|_{\HS} and Equation (6.20). The band version of the remainder bound follows from the uniform finite-list diagonal fixed at the start of the section. Its right side tends to zero as m\to\infty.

The spectral measures of D averaged with \tau have total mass tending to one, converge in moments to the standard Gaussian by Equation (6.21), and have uniformly bounded exponential moments at every fixed linear parameter by Equation (6.22). These facts imply weak convergence to the standard Gaussian and convergence of the integrals of every continuous function bounded by a polynomial times a fixed linear exponential. Here is a direct justification: the second moment gives tightness; the higher moment bounds give uniform integrability of each polynomial, so every subsequential limit has the Gaussian moments. Its exponential moments determine its characteristic function by its power series, hence determine the law. Using a larger exponential parameter makes the tails of any of the specified test functions uniformly small.

The scalar Hermite generating identity gives, with Z standard normal,

in squared mean. In fact its squared tail norm is e^{-2t}\sum_{j>m}(2t)^j/j!. Define the matrix multiplier

The just-proved convergence of spectral integrals, applied with each fixed m, yields

Together with Equations (6.27) and (6.20), first sending the sequence index and then m to infinity, this proves

There is no operator-norm bound on M in this argument; only its weighted trace moments are used.

Write \Psi(Y_0)=RJ_0, with J_0J_0^* equal to the support projection. On each block (R-R_*)^2\le\epsilon^2R_*^2, and R_* is scalar there. By cyclicity and positivity of trace,

Thus \Psi(Y_t) is close in squared mean to MR_*J_0. Its reduction is exactly

Finally Equation (6.3) transfers the approximation of these rectangular matrices to the approximation of their positive reduced square roots, proving Equation (6.25).□

Proof. Write a=1/L and b=1+1/L, so b-a=1. If r=\max(v,w)>0, discard the two denominator factors associated with the smaller eigenvalue and substitute q=Tr^2. This gives

For the identity in the middle, integration up to Q gives \log(1+bQ)-\log(1+aQ), which tends to \log(b/a). If v,w are positive and comparable as in the statement, let r_0=\min(v,w). Discard the two factors with coefficient a, and use b\ge1. The remaining integral is at most \int_0^\infty(1+Tr_0^2)^{-2}\,dT=r_0^{-2}. Since v+w\le3r_0, this proves the second bound.□

Proof. Let p_T(z\mid y) denote the density of the centered Gaussian with covariance C_T(y), and let p_T(z) be its marginal density. For each output z, the posterior density with respect to \eta is q_z(y)=p_T(z\mid y)/p_T(z). Applying 2\lambda H\le\Ical to this density and then averaging in z gives

Here and below the expectation on the right includes the observation noise. The posterior normalization does not depend on y and therefore does not contribute to the derivative.

We compute that derivative explicitly. For any positive covariance C and its derivative D_iC, differentiation of its determinant and inverse yields

Conditional on y, set B_i=C^{-1/2}(D_iC)C^{-1/2} and g=C^{-1/2}z. The vector g is standard Gaussian. Expanding its fourth moments gives \E(g^*B_ig-\tr B_i)^2=2\tr B_i^2; indeed the three Gaussian pairings give (\tr B_i)^2+2\tr B_i^2 before subtracting the mean. Consequently Equation (7.7) becomes

These calculations also apply to the Sobolev matrix field R. To see this without assuming additional regularity, fix T,L and the finite dimension of one member of the sequence. The map R\mapsto C_T has bounded derivative on positive semidefinite matrices, and its eigenvalues lie in [1,1+L]. For each fixed z, the map C\mapsto p(z\mid C)^{1/2} has bounded derivative on this covariance interval. The Sobolev chain rule therefore puts q_z^{1/2} in the form domain; p_T(z)>0 and is a harmless fixed normalization. The Sobolev approximation at the opening of Section 2 extends the logarithmic Sobolev inequality to this form-domain function. The squared derivatives integrate in z to the nonnegative finite expression in Equation (7.8). Approximation in the form domain and Tonelli thus justify both that equation and the subsequent scale integration. The finite-dimensional bounds used for form-domain membership do not enter its numerical coefficient.

In an eigenbasis of R, write v,w for the row and column eigenvalues. The covariance function is

Its divided difference, including its derivative at v=w, is

Multiplying its square by c_T(v)^{-1}c_T(w)^{-1} and integrating against dT/T^2 gives exactly the multiplier in Equation (7.4) for the squared entry |(D_iR)_{vw}|^2. When v=w=0, the covariance derivative is zero, which explains the value assigned to M_L(0,0).

It remains to obtain a bound independent of the fixed cap. For noncomparable pairs with at least one positive eigenvalue, Equation (7.2) gives entrywise

This includes a pair with exactly one zero eigenvalue. Their total expected derivative energy is therefore o(1). The kernel–kernel derivative is zero almost everywhere, as established in the Sylvester construction, and in any case has zero covariance multiplier. Lemma 7.1 now yields

Take the sequence limit at this fixed L, and use Equation (7.1). Stationarity makes the bound independent of t. This proves the proposition. In particular, the cap-dependent bound on noncomparable pairs is used only before its vanishing error has been removed.□

Proof. Direct differentiation gives

For v>0, substitution q=Tv^2 therefore shows

Discarding two of the three factors 1+q/L bounds K_L by the integral evaluated in Lemma 7.1, namely \log(1+L). If v\ge w\ge0, the fundamental theorem of calculus and Cauchy–Schwarz give

Integrating in T and using Tonelli proves the scalar bound K_L(v-w)^2. The value of the derivative integral at the single endpoint r=0 does not affect this argument, so it covers w=0.

Choose an orthonormal eigenbasis (e_a) of A with eigenvalues v_a and an orthonormal eigenbasis (f_b) of B with eigenvalues w_b. The entry of their functional-calculus difference between these two bases is

Multiply the scalar bound by |\langle e_a,f_b\rangle|^2 and sum over a,b. The sum on the right is K_L\|A-B\|_{\HS}^2, proving Equation (7.9). Finally, conditional on all data except G, Gaussian isometry gives

Apply the matrix bound and the squared-mean convergence in Proposition 6.8. The resulting factor K_L is fixed while the sequence limit is taken.□

Proof. Put q=1-\alpha^2. Given a posterior law \pi(dn) of N at fixed (Z,\mathcal D), the density at y after adding the OU noise is the integral of (2\pi q)^{-d/2}\exp(-|y-\alpha n|^2/(2q)) against \pi. Differentiating this kernel and dividing by its integral gives

The score of the standard Gaussian is -y. Subtracting it proves Equation (7.14). Kernel differentiation is justified first with N truncated; on compact positive time intervals its derivatives are bounded by Gaussian kernels times polynomials, and truncation can then be removed by conditional second moments.

The OU semigroup has generator \Delta-y\cdot D. For a positive posterior density r_u relative to \gamma, integration by parts gives

This is the entropy dissipation calculation of Lemma 2.3; positive-time smoothing, truncation of the density and spatial cutoffs justify it for finite-entropy posteriors. Averaging over (Z,\mathcal D) and integrating over \mathcal J bounds its nonnegative right side by the initial conditional information. Substitution of the score formula proves Equation (7.15).

For the last assertion, observation noise is independent of the initial data and diffusion path, and its law depends on them only through Y_t. Hence (\mathcal D,N)\longrightarrow Y_t\longrightarrow Z_T is a Markov chain. The chain rule and data processing give

These informations are finite for each member of the sequence: the covariance interval [\Id,(1+L)\Id] bounds the output entropy from above by that of a Gaussian of covariance (1+L)\Id and bounds its conditional entropy from below by that of a standard Gaussian. Thus \MI(Y_t;Z_T)\le(d/2)\log(1+L) in dimension d. This finiteness assertion is used only to justify the identities.□

Proof. For a block, consider the linear map \mathcal H:v\mapsto H[v] between its Euclidean coordinate space and its matrix space with the ordinary Hilbert–Schmidt norm. Its adjoint is

because H_i is real symmetric. Taking the adjoint in Equation (7.12) gives

This step has no factor involving the dimension of the block.

At fixed u,T, select precisely those positive blocks for which x=T\beta^2\in\mathcal X. On a selected block set

and set all other components, including those in the kernel, equal to zero. This vector is measurable with respect to (N_u,Z_T,\mathcal D). If m=\E[N-\alpha N_u\mid N_u,Z_T,\mathcal D], completing the square gives

The coordinates of distinct selected blocks are orthogonal. Their costs therefore add, and Equation (7.18) bounds the cost by C_1 times the sum of their unnormalized squared matrix norms. Their linear terms contract exactly to

Substitute these two statements in Lemma 7.4 and integrate in T.

For each realized positive block separately, the change of variable x=T\beta^2 has the exact Jacobian

Thus the block selection becomes x\in\mathcal X, and the coefficient of each block trace becomes precisely \beta^2. This proves Equation (7.17).

For completeness, all its polynomial terms are integrable on the compact parameter rectangle. The moment bounds of Lemma 6.7 control A_\beta and A_{u,\beta} under the weighted trace. Conditional on the initial data, diffusion path, and OU noise, B_{\beta,x} is a Gaussian matrix series whose vector covariance is at most (1+L)\Id. If its coefficients are K_j=\sum_i Q_{ij}H_i for a square root Q of that covariance, then for every vector v

The same Gaussian even-moment proof therefore gives, for every fixed integer p\ge1,

Trace Hölder bounds each mixed word by its factors’ moments. After multiplying by \beta^2 and summing, the right side is a constant times \tau(\Id), which is finite and bounded along the sequence. The measure \kappa(u)\,du\,dx/x^2 is finite on the chosen rectangle. Tonelli applies to the square terms, while these moment bounds justify Fubini for the linear terms. One can equivalently first truncate V in the conditional-mean inequality and then remove that truncation in this integrated squared norm.□

Proof. The flattening bound and the inverse substitution in Equation (7.19) give

by Lemma 7.3. All projected errors are summed before this comparison; no small block weight is divided out. Multiplication by the bounded weight \kappa(u) and integration in the compact interval \mathcal J preserve o(1).

To pass from this squared norm to arbitrary fixed products, include the parameter integrals in the finite tracial functional

Every matrix in the statement has bounded L^p(\mathcal T) norm for every fixed finite p. For B this was proved in Equation (7.20); for \widetilde B the same proof applies because its covariance is also at most (1+L)\Id. The remaining matrices are covered by the Gaussian moment lemma. For any q>p\ge2, interpolation with 1/p=\theta/2+(1-\theta)/q gives

For lower powers, Hölder and boundedness of \mathcal T(\Id) give the same conclusion. Expand the difference of two words of length r by replacing one factor at a time. Trace Hölder with all exponents r bounds each resulting trace by the product of its factors’ L^r(\mathcal T) norms. The changed factor tends to zero and the others remain bounded. There are only finitely many such terms. This proves the claim, including polynomial squared norms and linear terms with A-\alpha A_u.□

Proof. Suppose a continuous linear functional on L^p(\nu) annihilates all polynomials. Its representing density F belongs to L^{p'}(\nu), where p'=p/(p-1) and p'=\infty if p=1. Hölder and the hypothesis show that the finite signed measure F\,d\nu has exponential moments of every linear magnitude. For each \xi\in\R^k, the function

is entire: on every compact set of complex z, its derivatives are dominated by an integrable exponential. All its derivatives at zero vanish because (\xi\cdot w)^j is a polynomial. Its Taylor series and hence the entire function are zero. Evaluating at z=\mathrm i proves that the Fourier transform of F\,d\nu vanishes. Fourier uniqueness for finite measures can here be seen by convolving with a centered Gaussian: the convolution has zero Fourier transform and an integrable Fourier inversion formula, so is zero for every positive Gaussian variance; letting that variance tend to zero shows the original measure is zero. Thus F=0 almost everywhere. Duality, or equivalently separation of a proper closed subspace, proves the density assertion.□

Proof. First we prove the matrix comparison

All notation in this calculation is on one block. Define

Put q=g_x(A) and B=H[G]. The exact columnwise identity is

Indeed the bth column of J(qG)-qJ(G) is H_bqG-qH_bG. Conditional Gaussian averaging therefore bounds the square of the left side by

For the first and last terms in Equation (7.23), we used \|q\|_{\op}\le\sqrt{1+L} and the Gaussian isometry for the linear map D.

The function g_x is Lipschitz with a bound depending only on t,L, uniformly for all x>0. In fact, with a=\sqrt{8t} and r=xe^{av-4t}/L, differentiation gives

In an eigenbasis of A with eigenvalues r_a, the squared commutator norm is

This is a scalar Lipschitz estimate for matrix entries in the Hilbert–Schmidt norm. Averaging A=\sum_iN_iH_i over N gives

Multiply Equation (7.24) by the block weight, sum, and average. The total symmetry defect and Equation (7.13) prove Equation (7.22), uniformly in x at fixed t,L.

Every fixed higher moment of both sides of this comparison is bounded. Conditional on A and the initial data, H[g_x(A)G] is a Gaussian matrix series with coefficient-square sum at most (1+L)S, by the quadratic-form argument used in Equation (7.20). Also the Schatten ideal inequality gives

on each block. Interpolation and telescoping products, exactly as in Lemma 7.6, show that these two matrices may be interchanged in every fixed polynomial trace.

It remains to identify traces containing g_x(A)H[G]. Lemma 6.7 gives the joint scalar Gaussian moment law for A=H[N], H[G'], and H[G]. In particular, A_u=\alpha A+\sqrt{1-\alpha^2}H[G'] has the required joint law with them. To include the bounded continuous function g_x(A), fix x and a mixed word of length r. Choose a polynomial p approximating g_x in a sufficiently high finite L^q(\gamma_1) norm, where \gamma_1 is the scalar standard Gaussian law. Choose q larger than all the Hölder exponents needed for this fixed word, and take the approximation error at most 1; then \|p\|_{L^q(\gamma_1)} \le\sqrt{1+L}+1. Such approximations exist by the polynomial-density argument in Lemma 7.7, applied to one Gaussian coordinate. The scalar spectral measures of A under \tau converge to \gamma_1 by the moment and exponential-moment bounds in Lemma 6.7. Their higher moments are uniformly bounded. Consequently, for even q,

To justify this last passage directly, cut the scalar spectral variable off at |v|\le M, use weak convergence for the bounded continuous integrand there, and bound the complement by a higher uniform moment of A. Since g_x is bounded and p has fixed degree, that tail tends to zero as M\to\infty.

Replace each occurrence of g_x(A) by p(A). Telescoping the word and applying trace Hölder bounds the replacement error by a constant times the L^q error in Equation (7.25), with all other factors controlled by fixed higher moments. For factors p(A), the preceding L^q bound and moment convergence give a bound independent of how accurately p approximates g_x. Thus the error constant stays bounded as that approximation is improved. The remaining word is polynomial in the three independent Gaussian matrix series, so its trace has the scalar Gaussian limit. Let the scalar approximation error tend to zero. This proves the claimed limit at each fixed u,x; the scalar third coordinate is g_x(n)g.

Finally, the moment bounds for the original, bounded function g_x are uniform on the compact parameter rectangle, as are the bounds on polynomial coefficients in u,x and on \alpha. They dominate each mixed trace by an integrable constant. Dominated convergence thus permits parameter integration. This last argument uses the uniform bounds for g_x itself, and does not require choosing one polynomial approximant uniformly in x.□

Proof. Fix compact positive intervals \mathcal J,\mathcal X, and let n,g',g be independent scalar standard Gaussians. Write n_u=\alpha n+\sqrt{1-\alpha^2}\,g', and fix a scalar polynomial P(u,x,a,b). Give its matrix version any fixed ordering of the factors of each monomial. Lemmas 7.5, 7.6, and 7.8, in that order, imply

If the matrix polynomial is not symmetric, its prelimit cost is \tr(P^*P) as in Equation (7.17). The mixed trace law takes this cost to the real scalar P^2, so symmetry of the chosen ordering is unnecessary.

The sequence limit in Equation (7.27) has been taken for each fixed polynomial. We now optimize the tests in one squared-mean space containing the compact parameters. Give (u,x) the finite measure \kappa(u)\,du\,dx/x^2 on \mathcal J\times\mathcal X, independently of the scalar Gaussians n,g',g, and put

The first two coordinates of W are bounded. At each fixed u,x, the variable n_u is standard Gaussian, and |g_x(n)g|\le\sqrt{1+L}|g|. Cauchy–Schwarz therefore gives all linear exponential moments for its finite law, uniformly on the compact parameter rectangle. Lemma 7.7 shows that polynomials in the four coordinates of W are dense in this L^2 space.

Since \E[r^2\mid u,x]=1-\alpha^2\le1, the conditional mean m(W)=\E[r\mid W] belongs to the same space. For the finite joint measure just defined, completion of the square gives

Density makes the infimum of the final squared error zero, using single four-variable polynomials in the integrated space. Taking their supremum and then disintegrating with respect to (u,x) turns the right side of Equation (7.27) into

The conditional expectation here is taken at each fixed u,x. All approximation constants used before this optimization may depend on the fixed polynomial, t,L, and the compact intervals. The coefficient C_1 depends only on Equation (7.18).

We verify the scalar information identity including its endpoints. Fix x>0 and write z_x=g_x(n)g. Its information is finite: conditional variance lies in [1,1+L], so Gaussian entropy comparison gives

Indeed \E z_x^2\le1+L bounds its entropy above by \frac12\log(2\pi e(1+L)), while its conditional entropy is \frac12\log(2\pi e)+\E\log g_x(n) and is at least \frac12\log(2\pi e). Apply the scalar version of Lemma 7.4 and its entropy dissipation identity between times a and b, where 0<a<b<\infty. This gives

As a\downarrow0, the joint pair (n_a,z_x) converges in law to (n,z_x). Lower semicontinuity of relative entropy, applied to the joint law and the product of its marginals, gives \MI(n;z_x)\le\liminf_{a\downarrow0}\MI(n_a;z_x). The reverse inequality follows from the Markov chain z_x\longrightarrow n\longrightarrow n_a and data processing. Thus the initial endpoint is exactly \MI(n;z_x). At the other endpoint, the same chain gives

The equality follows by subtracting the conditional Gaussian entropy of variance 1-\alpha^2 from that of the standard Gaussian marginal. We have proved

Finally exhaust both positive parameter axes by compact intervals in Equation (7.29). Its integrand is nonnegative. Monotone convergence and Equation (7.31) prove Equation (7.26). In particular, the polynomial optimization and the exhaustion take place after the counterexample-sequence limit, so neither requires a uniform approximation over unbounded parameters.□

Proof. The Gaussian moment generating formula gives \E\ell=1. Let P_{x,n} be the conditional law of z_x given n, and let Q_x be its marginal law. Thus

Set p_0=\Prob\{|g|\ge\sqrt2\}>0. Suppose first that \ell\ge e^{2t}, and choose a scale with h=x\ell\in[h_0,t]. Since h\le t,

It follows that the event A_h=\{|z|\ge\sqrt h\} has conditional probability p=P_{x,n}(A_h)\ge p_0.

To bound its marginal probability, use an independent variable \ell' with the law of \ell. On x\ell'\le1, the conditional variance of the mixture component is at most 2, so its probability of A_h is at most 2e^{-h/4} by the Gaussian exponential bound. On the complementary event, Markov’s inequality and \E\ell'=1 give probability at most x. Hence

The last inequality holds for t\ge1 and h\le t, because te^{-2t}\le e^{-t/4}\le e^{-h/4}.

Apply data processing to the event indicator. The binary relative entropy is bounded below by

Choose once and for all

The preceding expression is then at least ch for h\in[h_0,t]. This entropy estimate is made separately for each conditional law P_{x,n}; the chosen event may therefore depend on n through h=x\ell.

Nonnegativity of relative entropy permits restricting the scale integral to these choices of n and x. Tonelli and the change of variable x=h/\ell yield

Under the probability change of measure with density \ell, completion of the square makes n a Gaussian of mean \sqrt{8t} and variance 1. Therefore \log\ell under that measure is Gaussian of mean 4t and variance 8t. If \Phi is the standard Gaussian distribution function, then

This proves both inequalities in Equation (7.32).□

Proof of Theorem 1.1. Suppose that the asserted dimension-free bound fails. Lemma 3.1 gives compact centered counterexamples with normalized logarithmic Sobolev constant 1 and linear subgaussian parameter tending to zero. Proposition 3.8 shows that their positive-time regularized constants converge to 1. Proposition 3.9 supplies the stationary laws and near-eigenfunction fields at the fixed regularization time s=1/8, in either the strict entropy branch or the spectral gap branch. Proposition 4.3 gives the exact prediction curve in both cases. Proposition 5.1 then gives the symmetric tensor hierarchy, and Proposition 6.8 constructs its matrix square root and frozen block description. These are precisely the inputs used in the present section.

For each fixed t>0 and L\ge1, combine Propositions 7.2 and 7.9. They give

where C_0C_1 is independent of both fixed parameters. Choose an arbitrarily large but finite t\ge h_0 and the fixed cap L=t. Lemma 7.10 makes the left side at least (c/2)\log(t/h_0), which exceeds C_0C_1 for a sufficiently large choice of t. This is a contradiction.

All counterexample limits used to obtain Equation (7.34) were taken with this t,L fixed. Only afterward was their numerical size chosen. Thus no uniform estimate on the intermediate errors as t,L\to\infty is needed. The contradiction excludes an unbounded ratio R(\mu)/a^2 over the measures in the theorem. Finally the convention \Ent_\mu(f^2)\le2R(\mu)\int|Df|^2\,d\mu absorbs the factor 2 into a universal constant and gives the asserted inequality for every smooth compactly supported real test function.□