Lyapunov–Stein duality - The KLS theorem and its methods

Part of the fixed-cut archive, Chapter The fixed cut: the mass martingale and the Carleson estimate; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 30.3 as Lemma D13.1. At every time $t>0$ , $s_t\inner{K_t}{\mathscr L_{A_t}^{-1}K_t}\le4/t$ ; the lemma also gives the cut-oriented source scale and direct-sum invariance of $\lambda_{\rm cut}$ . The proof combines the anisotropic Brascamp–Lieb quadratic-form estimate from Lemma 30.2 with finite-dimensional Hilbert-space duality for the Lyapunov operator. The proof is unconditional for $t>0$ and asserts nothing at $t=0$ .

Support convention: $K_t$ lives on the range of $A_t$ ((D13.2)). There the support inverse coincides with the pseudoinverse (D13.3).
The two-color pairing (D13.10), followed by Cauchy–Schwarz and Brascamp–Lieb on the $t$ -uniformly log-concave posterior, gives (D13.11) for every symmetric test matrix $M$ .
The right side of step 2 is written as $\inner M{\mathscr L_{A_t}M}$ ((D13.12)). The duality identity (D13.13), taken at $M=\mathscr L_{A_t}^{-1}K_t$ , then gives (D13.5). Multiplying by $\lambda_{\rm cut}$ gives (D13.7).
Block-diagonal invariance of $\mathscr L_{A\oplus B}$ proves (D13.8).
An auxiliary remark, which is not part of the lemma (Remark D13.1), gives the harmonic-mean formula (D13.17) and the two-tail calibration (D13.20).

Setup and covariance-support convention. Fix a time $t>0$ in the stochastic-localization process and a cut $E$ for which $p_t,q_t>0$ . Retain the manuscript notation

A_t=\Cov_{\mu_t}(X),\qquad s_t=p_tq_t, \qquad K_t=G_t+(q_t-p_t)\delta_t\delta_t^T.

(D13.1)

Let $H_t=\operatorname{Ran}(A_t)$ and let $P_t$ be the orthogonal projection onto $H_t$ . If $v\in\ker A_t$ , then $\E_{\mu_t}(v\cdot(X-a_t))^2=0$ . Hence $X-a_t\in H_t$ almost surely, and the conditional means and covariance matrices defining $K_t$ also live on $H_t$ . In particular,

K_t=P_tK_tP_t.

(D13.2)

For a positive-semidefinite matrix $A$ , write $H=\operatorname{Ran}(A)$ . When it is applied to a supported matrix $K=PKP$ , the notation $\mathscr L_A^{-1}K$ below means the inverse on the Hilbert space $\mathrm{Sym}(H)$ , followed by zero extension to the ambient space. On such inputs this agrees with the ambient Moore–Penrose inverse $\mathscr L_A^\dagger$ . Indeed, in an $A$ -eigenbasis with eigenvalues $\lambda_i\ge0$ , the latter is

(\mathscr L_A^\dagger C)_{ij} = \begin{cases} \displaystyle\frac{2C_{ij}}{\lambda_i+\lambda_j},&\lambda_i+\lambda_j>0,\\ 0,&\lambda_i+\lambda_j=0. \end{cases}

(D13.3)

For a covariance contrast $K=PKP$ , this is exactly the inverse on $\mathrm{Sym}(H)$ and is independent of the ambient zero extension.

Lemma D13.1 (= Lemma 30.3)

On symmetric matrices over $H_t$ , define

\mathscr L_{A_t}(M)=\frac{A_tM+MA_t}{2}.

(D13.4)

Then

s_t\inner{K_t}{\mathscr L_{A_t}^{-1}K_t}\le\frac4t.

(D13.5)

For $K\ne0$ set

\lambda_{\rm cut}(A,K) :=\frac{\norm K_{\HS}^2}{\inner K{\mathscr L_A^{-1}K}},

(D13.6)

and set $\lambda_{\rm cut}(A,0)=0$ . Then

s_t\norm{K_t}_{\HS}^2 \le\frac{4\lambda_{\rm cut}(A_t,K_t)}t.

(D13.7)

Moreover, if $A,B\succeq0$ and $K$ is supported on $\operatorname{Ran}(A)$ , then

\lambda_{\rm cut}(A\oplus B,K\oplus0)=\lambda_{\rm cut}(A,K).

(D13.8)

Proof D13.1

We first retain the anisotropic quadratic-form estimate inside the certified proof of Lemma 30.2. Let $M\in\mathrm{Sym}(H_t)$ and set

f_M(x)=(x-a_t)^TM(x-a_t)-\Tr(MA_t), \qquad g=\frac{\one_E-p_t}{\sqrt{s_t}}.

(D13.9)

Then $\E_{\mu_t}g=\E_{\mu_t}f_M=0$ , $\E_{\mu_t}g^2=1$ , and the two-color covariance identity gives

\E_{\mu_t}[g f_M]=\sqrt{s_t}\,\inner{K_t}{M}.

(D13.10)

The posterior $\mu_t$ , considered on its affine support $a_t+H_t$ , is $t$ -uniformly log-concave. Cauchy–Schwarz and Brascamp–Lieb on that support Brascamp & Lieb, 1976 therefore give

\begin{aligned} s_t\inner{K_t}{M}^2 &\le \Var_{\mu_t}(f_M)\\ &\le \frac1t\E_{\mu_t}\abs{\nabla_{H_t}f_M}^2 =\frac4t\Tr(MA_tM). \end{aligned}

(D13.11)

The quadratic test belongs to the Brascamp–Lieb form domain because the finite-time posterior has the Gaussian factor $e^{-t\abs{x}^2/2}$ on its support; alternatively one obtains (D13.11) from smooth truncations and form closure. Thus no boundedness of $f_M$ is being assumed.

We now express the right side of (D13.11) in its exact Hilbert-space form. On $\mathrm{Sym}(H_t)$ , equipped with the Hilbert–Schmidt inner product, $\mathscr L_{A_t}$ is self-adjoint and positive definite, and

\inner{M}{\mathscr L_{A_t}M} =\Tr(MA_tM) =\norm{A_t^{1/2}M}_{\HS}^2.

(D13.12)

For any positive-definite self-adjoint operator $L$ on a finite-dimensional real Hilbert space, Cauchy–Schwarz after inserting $L^{\pm1/2}$ yields the duality identity

\sup_{M\ne0}\frac{\inner{K}{M}^2}{\inner{M}{LM}} =\inner K{L^{-1}K}.

(D13.13)

More explicitly, $\inner KM=\inner{L^{-1/2}K}{L^{1/2}M}$ gives “ $\le$ ”, and equality is attained at $M=L^{-1}K$ when $K\ne0$ ; for $K=0$ both sides vanish.

Apply (D13.13) to $L=\mathscr L_{A_t}$ in (D13.11). Equivalently, insert $M=\mathscr L_{A_t}^{-1}K_t$ . If $K_t\ne0$ , positivity gives $d_t:=\inner{K_t}{\mathscr L_{A_t}^{-1}K_t}>0$ , and

s_t d_t^2\le\frac4t d_t.

(D13.14)

Division by $d_t$ proves (D13.5). If $K_t=0$ , that inequality is simply $0\le4/t$ . In the nonzero case, multiplication by $\norm{K_t}_{\HS}^2/d_t=\lambda_{\rm cut}(A_t,K_t)$ gives (D13.7). In the zero case, both sides of that source-scale inequality are zero by the separate convention $\lambda_{\rm cut}(A_t,0)=0$ .

It remains to verify the direct-sum claim. The block-diagonal subspace is invariant under $\mathscr L_{A\oplus B}$ , and the support inverse satisfies

\mathscr L_{A\oplus B}^{-1}(K\oplus0) =\mathscr L_A^{-1}K\oplus0.

(D13.15)

Both the Hilbert–Schmidt numerator and the Lyapunov denominator in the definition of $\lambda_{\rm cut}$ are therefore unchanged, proving (D13.8); the $K=0$ case again follows from the separate convention.

Remark D13.1 (Auxiliary eigenbasis formula and two-tail calibration)

The following identities are useful calibrations, but they are not assertions of Lemma D13.1. Let $A\succeq0$ , let $H=\operatorname{Ran}(A)$ , and let $K\ne0$ be supported on $H$ . In an eigenbasis of $A|_H$ with positive eigenvalues $\lambda_1,\ldots,\lambda_r$ , formula (D13.3) gives

\inner K{\mathscr L_A^{-1}K} =\sum_{i,j=1}^r\frac{2\abs{K_{ij}}^2}{\lambda_i+\lambda_j} =\sum_{i,j=1}^r \frac{\abs{K_{ij}}^2}{(\lambda_i+\lambda_j)/2}.

(D13.16)

Since $\norm K_{\HS}^2=\sum_{i,j=1}^r\abs{K_{ij}}^2$ , one obtains the auxiliary identity

\lambda_{\rm cut}(A,K) =\frac{\displaystyle\sum_{i,j=1}^r\abs{K_{ij}}^2} {\displaystyle\sum_{i,j=1}^r \frac{\abs{K_{ij}}^2}{(\lambda_i+\lambda_j)/2}}.

(D13.17)

Thus $\lambda_{\rm cut}(A,K)$ is the $\abs{K_{ij}}^2$ -weighted harmonic mean of $(\lambda_i+\lambda_j)/2$ .

For the anisotropic two-tail pair of Proposition 25.2,

A_\Lambda=\diag(\Lambda,1,\ldots,1), \qquad K_\Lambda=8a\varphi(a)\Lambda\,e_1e_1^T.

(D13.18)

Since $\mathscr L_{A_\Lambda}(e_1e_1^T)=\Lambda e_1e_1^T$ , one has $\mathscr L_{A_\Lambda}^{-1}K_\Lambda=K_\Lambda/\Lambda$ . Hence

\lambda_{\rm cut}(A_\Lambda,K_\Lambda) =\frac{\norm{K_\Lambda}_{\HS}^2} {\norm{K_\Lambda}_{\HS}^2/\Lambda} =\Lambda,

(D13.19)

or equivalently

\lambda_{\rm cut}(A_\Lambda,K_\Lambda)=\Lambda.

(D13.20)

This auxiliary calibration is why prop:two-tail remains in the dossier’s declared dependencies even though the formal lemma itself ends with (D13.8).

Scope, hypotheses, and initial-time exclusion. The proof uses only $t>0$ , the finite-time posterior Brascamp–Lieb inequality on the covariance support, the two-color identity already contained in the certified dependency Lemma 30.2, and finite-dimensional Hilbert-space duality. It is unconditional in the manuscript’s localization setup and has no unclosed analytic step. It makes no assertion at $t=0$ : the factor $t^{-1}$ is singular, and the pointwise estimate supplies neither an integrable initial-time bound nor an expected covariance-occupation theorem.

Obstructions respected. The ledger node has no formal bounded_by edge. The formal statement nonetheless remains on the safe side of the known route fences: it uses the full cut-oriented tensor rather than only radial or projection data, and direct-sum invariance removes only genuinely irrelevant spectator blocks. The auxiliary calibration records $\lambda_{\rm cut}(A_\Lambda,K_\Lambda)=\Lambda$ , not a false dimension-free scale, on the anisotropic two-tail obstruction. No operator-to-trace upgrade or high-rank occupation estimate is claimed.

References¶

Brascamp, H. J., & Lieb, E. H. (1976). On Extensions of the Brunn–Minkowski and Prékopa–Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation. Journal of Functional Analysis, 22(4), 366–389. 10.1016/0022-1236(76)90004-5