The posterior eigenfunction defect - The KLS theorem and its methods

Part of the fixed-eigenfunction mechanism, Chapter The fixed eigenfunction: following one eigenfunction through localization; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 21.1 as Theorem D14.1. The setting is a smooth, strongly log-concave, isotropic law, a normalized first eigenfunction $f$ , and the planted Gaussian observation channel. The posterior defect $R_t$ obeys three exact integration-by-parts identities (D14.7), two variance budgets (D14.8) and an averaged gradient bound (D14.9). The identities come from pairing a posterior generator relation with constant, linear and quadratic tests. The budgets come from filtering martingales and the planted cancellation $R_t(X)=B_t^{\mathrm{obs}}\cdot\nabla f(X)$ . The imported node Theorem 25.1 is not used.

The posterior density (D14.10) and the innovation Brownian motion give the filtering equation (D14.13). The posterior generator is (D14.16), with its domains justified by cutoffs and stopping.
The generator relation (D14.17) is paired with constants, coordinates and centered quadratics $Q_D$ . This gives the three identities (D14.7).
The planted cancellation (D14.22) gives $\E\E_tR_t^2=t\lambda$ ((D14.24)). Combined with the Itô isometry for $m_t$ and the first identity of step 2, this yields the centered-defect budget.
Filtering applied to $\nabla f$ gives the isometry (D14.28). Conditional Jensen, together with $b_0=\lambda g_0$ from step 2, bounds the $C_t$ budget.
The gradient of $R_t$ is evaluated at the planted point ((D14.30)). The weighted Bochner identity and convexity bound $\E_\mu\norm{\nabla^2f}_{\HS}^2\le\lambda^2$ , which proves (D14.9).

Refined statement. The regular setting in the following theorem makes explicit the domain and normalization conventions implicit in Lemma 21.1. For vectors $v,w\in\R^n$ we use $(v\otimes w)_{ij}=v_iw_j$ , and $\operatorname{sym}M=(M+M^T)/2$ .

Theorem D14.1 (= Lemma 21.1)

Let

\dd\mu(x)=Z^{-1}e^{-V(x)}\dd x

(D14.1)

be a smooth, strongly log-concave, centered isotropic probability measure on $\R^n$ : $V\in C^\infty(\R^n)$ and $\nabla^2V\succeq\varepsilon I_n$ for some $\varepsilon>0$ . Let $L=\Delta-\nabla V\cdot\nabla$ denote its Friedrichs self-adjoint realization in $L^2(\mu)$ , and let $f$ be a normalized first nonconstant eigenfunction,

-Lf=\lambda f, \qquad \E_\mu f=0, \qquad \E_\mu f^2=1.

(D14.2)

In particular, $f$ is understood in the generator domain; all identities below use the closed Dirichlet-form interpretation, so there is no boundary condition hidden at infinity.

Take $X\sim\mu$ and an independent standard Brownian motion $B^{\mathrm{obs}}$ , set

c_t=tX+B_t^{\mathrm{obs}}, \qquad \mathcal F_t^{\mathrm{obs}}=\sigma(c_s:0\le s\le t),

(D14.3)

and let $\mu_t=\mathcal L(X\mid\mathcal F_t^{\mathrm{obs}})$ . Write $\E_t$ for integration against $\mu_t$ , $a_t=\E_tX$ , and

m_t=\E_tf, \qquad g_t=\E_t[(f-m_t)(X-a_t)], \qquad H_t=\E_t[(f-m_t)(X-a_t)^{\otimes2}].

(D14.4)

For the posterior defect

R_t(x)=(c_t-tx)\cdot\nabla f(x), \qquad \rho_t=R_t-\E_tR_t,

(D14.5)

define, with the displayed tensor orientation,

b_t=\E_t\nabla f, \qquad C_t=\E_t[(X-a_t)\otimes\nabla f], \qquad u_t=\E_t[\rho_t(X-a_t)], \qquad K_t=\E_t[\rho_t(X-a_t)^{\otimes2}].

(D14.6)

Then, for every $t\ge0$ , almost surely in the pointwise posterior identities,

\begin{aligned} \E_tR_t&=\lambda m_t, &\lambda g_t&=b_t+u_t, &\lambda H_t&=2\operatorname{sym}C_t+K_t, \end{aligned}

(D14.7)

\begin{aligned} \E\Var_t(R_t) &=t\lambda-\lambda^2\int_0^t\E\abs{g_s}^2\dd s, &\E\int_0^t\norm{C_s}_{\HS}^2\dd s &\le \lambda-\lambda^2\abs{g_0}^2, \end{aligned}

(D14.8)

\begin{aligned} \E\E_t\abs{\nabla R_t}^2 &\le t\lambda^2+t^2\lambda. \end{aligned}

(D14.9)

Here the un-subscripted expectation averages the planted observation model, and $\Var_t(R_t)=\E_t\rho_t^2$ .

Proof D14.1

Posterior and innovation equations. The likelihood of the observation path up to time $t$ , conditional on $X=x$ , depends on that path only through $c_t$ and is proportional to $\exp(c_t\cdot x-t\abs{x}^2/2)$ . Hence

\dd\mu_t(x) =\frac{\exp(c_t\cdot x-t\abs{x}^2/2)} {\int\exp(c_t\cdot y-t\abs{y}^2/2)\dd\mu(y)}\,\dd\mu(x).

(D14.10)

Put

W_t=c_t-\int_0^ta_s\dd s.

(D14.11)

Since $\E[X\mid\mathcal F_t^{\mathrm{obs}}]=a_t$ , the process $W$ is a continuous $\mathcal F_t^{\mathrm{obs}}$ -local martingale with quadratic covariation $[W^i,W^j]_t=\delta_{ij}t$ . Lévy’s characterization therefore makes $W$ an $n$ -dimensional Brownian motion. Applying Itô’s formula to (D14.10) gives

\dd\mu_t(x)=\mu_t(x)(x-a_t)\cdot\dd W_t.

(D14.12)

Consequently, for every fixed scalar $\phi\in L^2(\mu)$ ,

\dd\E_t\phi=\Cov_t(\phi,X)\cdot\dd W_t.

(D14.13)

We record the localization and domain justification used throughout. First take bounded smooth truncations of $\phi$ and stop at the first time when the stochastic-integral quadratic variation or the posterior moments occurring in the calculation reach level $N$ . Equations (D14.12)–(D14.13) are then ordinary bounded Itô identities. Conditional expectation is a contraction in $L^2$ , so bounded truncations of $f$ and of each component of $\nabla f$ converge to their square-integrable conditional-expectation martingales. Their stopped stochastic integrals therefore converge in $L^2$ , and the stopping times increase to infinity. This removes the stopping without losing any equality below.

For the spatial integrations by parts, the posterior potential is

V_t(x)=V(x)-c_t\cdot x+\frac t2\abs{x}^2, \qquad \nabla^2V_t\succeq(\varepsilon+t)I_n.

(D14.14)

Thus $\mu_t$ has all polynomial moments. The closed Dirichlet form

\mathcal E_t(h,k)=\E_t[\nabla h\cdot\nabla k], \qquad h,k\in W^{1,2}(\mu_t),

(D14.15)

has generator

L_t=\Delta-\nabla V_t\cdot\nabla =L+(c_t-tx)\cdot\nabla.

(D14.16)

The identities below are first paired with compactly supported cutoffs of the coordinate and centered-quadratic tests. The Gaussian tails of $\mu_t$ , the $L^2$ bounds established by the planted coupling below, and closedness of $\mathcal E_t$ let the cutoff radius tend to infinity. Equivalently, the coordinates and centered quadratics belong to the required local form domain, and $f$ belongs to the maximal generator domain of $L_t$ because both $f$ and $L_tf$ are in $L^2(\mu_t)$ . This specifies the domain in every use of posterior integration by parts.

The defect identities. The eigenfunction equation and (D14.16) give the exact generator relation

-L_tf=\lambda f-R_t.

(D14.17)

Pairing this identity with the constant function in the posterior Dirichlet form yields

0=\E_t[-L_tf]=\lambda m_t-\E_tR_t,

(D14.18)

which is the first identity in (D14.7).

Pair next with the coordinate $x_i-a_{t,i}$ . Posterior integration by parts gives

\E_t[(-L_tf)(X_i-a_{t,i})]=\E_t\partial_i f=(b_t)_i.

(D14.19)

Because $\E_t(X-a_t)=0$ and $\rho_t=R_t-\lambda m_t$ , the same left-hand side, evaluated with (D14.17), is $(\lambda g_t-u_t)_i$ . Thus $\lambda g_t=b_t+u_t$ .

Finally fix an arbitrary symmetric matrix $D$ and set

Q_D(x)=(x-a_t)^TD(x-a_t)-\Tr(DA_t).

(D14.20)

Then $\E_tQ_D=0$ , $\nabla Q_D=2D(x-a_t)$ , and our orientation for $C_t$ gives

\begin{aligned} \E_t[(-L_tf)Q_D] &=\E_t[\nabla f\cdot\nabla Q_D] =2\langle D,\operatorname{sym}C_t\rangle_{\HS},\\ \E_t[(\lambda f-R_t)Q_D] &=\langle D,\lambda H_t-K_t\rangle_{\HS}. \end{aligned}

(D14.21)

The two expressions agree by (D14.17). Since they agree for every symmetric $D$ and both $H_t,K_t$ are symmetric, we obtain $\lambda H_t=2\operatorname{sym}C_t+K_t$ .

The centered-defect budget. At the planted point, the definition of the observation gives the cancellation

R_t(X)=B_t^{\mathrm{obs}}\cdot\nabla f(X).

(D14.22)

The usual energy identity for the normalized eigenfunction is

\E_\mu\abs{\nabla f}^2 =\langle f,-Lf\rangle_{L^2(\mu)}=\lambda.

(D14.23)

Since $B_t^{\mathrm{obs}}$ is independent of $X$ , has mean zero, and has covariance $tI_n$ , the conditional-law property and (D14.22) imply

\E\E_tR_t^2 =\E\abs{B_t^{\mathrm{obs}}\cdot\nabla f(X)}^2 =t\lambda.

(D14.24)

In particular, the $L^2(\mu_t)$ assertion used in the domain paragraph holds for almost every observation.

Apply (D14.13) to $f$ . Since $m_0=0$ and $\Cov_t(f,X)=g_t$ , the square-integrable martingale $m$ satisfies

\dd m_t=g_t\cdot\dd W_t, \qquad \E m_t^2=\int_0^t\E\abs{g_s}^2\dd s.

(D14.25)

The second equality is first the stopped Itô isometry; it remains an equality after removal because $m_t=\E[f(X)\mid\mathcal F_t^{\mathrm{obs}}]$ is bounded in $L^2$ . Using $\E_tR_t=\lambda m_t$ and conditional variance decomposition in (D14.24) now gives

\E\Var_t(R_t) =\E\E_tR_t^2-\E(\E_tR_t)^2 =t\lambda-\lambda^2\int_0^t\E\abs{g_s}^2\dd s.

(D14.26)

The $C_t$ martingale budget. Apply the vector form of (D14.13) to the fixed vector field $\nabla f$ . With the orientation specified in the theorem,

\dd b_t=C_t^T\dd W_t.

(D14.27)

The martingale $b_t=\E[\nabla f(X)\mid\mathcal F_t^{\mathrm{obs}}]$ is square-integrable by (D14.23). The stopped vector Itô isometry and the same $L^2$ stopping-removal argument therefore give

\E\int_0^t\norm{C_s}_{\HS}^2\dd s =\E\abs{b_t}^2-\abs{b_0}^2.

(D14.28)

Conditional Jensen and the tower property show that $\E\abs{b_t}^2\le\E\abs{\nabla f(X)}^2=\lambda$ . At time zero, $c_0=0$ and hence $R_0=\rho_0=u_0=0$ ; the already proved coordinate identity gives $b_0=\lambda g_0$ . Substitution in (D14.28) proves the second estimate in (D14.8).

The averaged gradient-defect budget. Differentiating $R_t$ with respect to its spatial argument gives

\nabla R_t(x)=\nabla^2f(x)(c_t-tx)-t\nabla f(x).

(D14.29)

Evaluate at the planted point and use $c_t-tX=B_t^{\mathrm{obs}}$ . Independence and centering of $B_t^{\mathrm{obs}}$ make the cross term vanish, while its covariance is $tI_n$ . Therefore

\E\E_t\abs{\nabla R_t}^2 =t\E_\mu\norm{\nabla^2f}_{\HS}^2+t^2\E_\mu\abs{\nabla f}^2.

(D14.30)

For the Friedrichs realization, the integrated weighted Bochner identity, justified by the same compact cutoff approximation, is

\E_\mu(Lf)^2 =\E_\mu\norm{\nabla^2f}_{\HS}^2 +\E_\mu\langle\nabla^2V\nabla f,\nabla f\rangle.

(D14.31)

Convexity of $V$ , the eigenfunction equation, and normalization consequently imply

\E_\mu\norm{\nabla^2f}_{\HS}^2 \le\E_\mu(Lf)^2=\lambda^2.

(D14.32)

Combining this with (D14.23) and (D14.30) gives (D14.9). All limiting operations used here are either $L^2$ limits of the two conditional-expectation martingales or closed-form $L^2$ limits of the spatial cutoff approximations, so the auxiliary stopping and spatial cutoffs have now been removed completely.

Obstructions respected. The ledger node has no formal bounded_by edge. The route fences were nevertheless checked individually. The proof uses no cut or slice estimate, so rem:two-tail-slice-bounds, rem:profile-circularity, and rem:single-coordinate-cuts are not engaged. It derives exact coordinate and full symmetric-matrix integration-by-parts identities rather than a dimension-free quadratic-chaos theorem from radial or projection tests, so it does not cross rem:projection-ceiling. It uses neither the crude covariance integral nor a relative covariance occupation bound, respecting rem:crude-insufficient and rem:relative-ceiling. Finally, it never bounds a posterior covariance operator norm, never unwhitens a tensor, and never invokes the refuted truncated-exponential variable-weight Stein shortcut; hence it also respects the covariance-spike and spectral-unwhitening warnings. The transport and needle warnings are inapplicable because neither mechanism occurs.