Proof of The Gaussian Relative Score Paired with the Gradient Field of a Diagonal Quadratic Profile

Each result cited is universally quantified over the data in its own statement.

Throughout, $x_{k}=\langle x,e_{k}\rangle$ is the $k$-th coordinate of $x\in X$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates), integrals are taken on the measure space $(X,\mathcal{B}(X),\nu)$, and $L^{2}(\nu)$, with inner product $\langle\cdot,\cdot\rangle_{L^{2}(\nu)}$ and norm $\lVert\cdot\rVert_{L^{2}(\nu)}$, is as in the statement. Since $\nu\in\mathcal{P}^{a}_{\rho}$, we have $\nu\in\mathcal{P}_{2}(X)$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments, that is, $\int_{X}|x|^{2}\,\nu(dx)<\infty$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space. The variances $c_{k}$ are positive by Variance Sequences and Their Truncations §variances, $c$ being a variance sequence by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian.

Step 1 (the coordinates are square-integrable). Fix $k\in\mathbb{N}$. The function $x\mapsto x_{k}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, so $x\mapsto x_{k}^{2}$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $x\in X$, since $e_{k}$ is a unit vector of the orthonormal basis, The Cauchy-Schwarz Inequality in a Real Inner Product Space gives $|x_{k}|=|\langle x,e_{k}\rangle|\le|x|\,|e_{k}|=|x|$, hence $x_{k}^{2}\le|x|^{2}$. By monotonicity of the integral of nonnegative measurable functions (Linearity and Monotonicity of the Lebesgue Integral §nonnegative),

$$\int_{X}x_{k}^{2}\,\nu(dx)\le\int_{X}|x|^{2}\,\nu(dx)<\infty .$$

The power $|x_{k}|^{2}$ with the real exponent $2$ used in Power-Integrable Functions and the p-Seminorm §space is the natural power $|x_{k}|\,|x_{k}|=x_{k}^{2}$ by Properties of Real Powers of Nonnegative Real Numbers §agreement. Hence $x\mapsto x_{k}$ is $2$-integrable with respect to $\nu$ in the sense of Power-Integrable Functions and the p-Seminorm §space, so its class lies in $L^{2}(\nu)$, and $x\mapsto x_{k}^{2}$ is integrable by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral; we write $m_{k}=\int_{X}x_{k}^{2}\,\nu(dx)$, a real number. This is the first assertion of claim 1.

Step 2 (a cut-off of the identity map of $\mathbb{R}$). Apply Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball with $q=1$, identifying $\mathbb{R}^{1}$ with $\mathbb{R}$, so that the Euclidean norm of $t$ is $|t|$. Let $\chi$, $\chi_{R}$ and $\psi_{R}$ be the functions of that lemma, and let $M\ge0$ be the constant of its clause 2, which does not depend on $R$. For $m\in\mathbb{N}$, which is positive, put $g_{m}=\partial_{1}\psi_{m}:\mathbb{R}\to\mathbb{R}$. The following hold for every $m\in\mathbb{N}$.

(a) By clause 2 of that lemma $\psi_{m}$ is smooth, hence of class $C^{2}$ on $\mathbb{R}^{1}$ by Smooth Map on a Euclidean Open Set, so by clause 2 of C^k Maps on a Euclidean Open Set (with $k=1$ there) $g_{m}$ is of class $C^{1}$ on $\mathbb{R}^{1}$, and its partial derivative, written $g_{m}'=\partial_{1}g_{m}$, is the iterated partial derivative $\partial_{1}\partial_{1}\psi_{m}$ of clause 4 there. By clause 1 there, $g_{m}$ and $g_{m}'$ are continuous at every point of $\mathbb{R}$.

(b) By clause 2 of that lemma, $|g_{m}(t)|\le M|t|$ and $|g_{m}'(t)|\le M$ for every $t\in\mathbb{R}$.

(c) By clause 3 of that lemma, $g_{m}(t)=t$ and $g_{m}'(t)=\delta_{11}=1$ for every $t\in\mathbb{R}$ with $|t|<m$.

(d) $g_{m}$ is bounded. If $|t|\le2m$ then $|g_{m}(t)|\le2mM$ by (b). If $|t|>2m$, put $\eta=|t|-2m>0$; by clause 1 of that lemma, $\chi_{m}(u)=0$ and hence $\psi_{m}(u)=\tfrac12\chi_{m}(u)u^{2}=0$ for every $u\in\mathbb{R}$ with $|u|\ge2m$, and for $0<|s|<\eta$ we have $|t+s|\ge|t|-|s|>2m$, so the difference quotient $(\psi_{m}(t+s)-\psi_{m}(t))/s$ is $0$. By Partial Derivative on a Euclidean Open Set the value $g_{m}(t)=\partial_{1}\psi_{m}(t)$ therefore satisfies $|g_{m}(t)|<\varepsilon$ for every $\varepsilon>0$, so $g_{m}(t)=0$. Thus $|g_{m}(t)|\le2mM$ for every $t\in\mathbb{R}$.

Step 3 (cylindrical approximants of the coordinate). Fix $k,m\in\mathbb{N}$ and define $h_{m}:\mathbb{R}^{k}\to\mathbb{R}$ by $h_{m}(y)=g_{m}(y_{k})$. For $i\in[k]$ with $i\ne k$, varying the $i$-th variable of $y$ does not change $y_{k}$, so every difference quotient of $h_{m}$ at $y$ in the $i$-th variable is $0$ and, by Partial Derivative on a Euclidean Open Set, $\partial_{i}h_{m}(y)$ exists and equals $0$. In the $k$-th variable the difference quotient of $h_{m}$ at $y$ with increment $s$ is $(g_{m}(y_{k}+s)-g_{m}(y_{k}))/s$, the difference quotient of $g_{m}$ at $y_{k}$, so by the same definition $\partial_{k}h_{m}(y)$ exists and equals $g_{m}'(y_{k})$. Since $|y_{k}-z_{k}|\le\lVert y-z\rVert$ for $y,z\in\mathbb{R}^{k}$, the continuity of $g_{m}$ and $g_{m}'$ at $y_{k}$ (Step 2(a)) gives the continuity of $h_{m}$ and of $y\mapsto g_{m}'(y_{k})$ at $y$; constant functions are continuous. Hence $h_{m}$ is of class $C^{1}$ on $\mathbb{R}^{k}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set, and $h_{m}$ and all its partial derivatives are bounded by Step 2(b) and (d). Thus $h_{m}\in C^{1}_{b}(\mathbb{R}^{k})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded. Consequently the function

$$\varphi_{m}=h_{m}\circ p_{k},\qquad\varphi_{m}(x)=g_{m}(x_{k}),$$

is a bounded $C^{1}$ cylindrical function by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, with representation $(k,h_{m})$, and by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial its $k$-th partial derivative is $\partial_{k}\varphi_{m}=(\partial_{k}h_{m})\circ p_{k}$, that is, $\partial_{k}\varphi_{m}(x)=g_{m}'(x_{k})$ for $x\in X$.

Step 4 (the score identity along the approximants). By The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score applied to $\varphi_{m}$, together with the integrability of $x\mapsto x_{k}\varphi_{m}(x)$ (Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §coordinate-integrable) and of $\partial_{k}\varphi_{m}$ (Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable) and linearity of the integral (Linearity and Monotonicity of the Lebesgue Integral §integrable),

$$\langle\zeta_{k},\varphi_{m}\rangle_{L^{2}(\nu)}=\frac{1}{c_{k}}\int_{X}x_{k}\,g_{m}(x_{k})\,\nu(dx)-\int_{X}g_{m}'(x_{k})\,\nu(dx)\qquad(m\in\mathbb{N}).\tag{1}$$

Step 5 (dominated convergence). Fix $x\in X$. By The Archimedean Property of the Real Numbers there is $m_{0}\in\mathbb{N}$ with $|x_{k}|<m_{0}$, so $|x_{k}|<m$ for every $m\ge m_{0}$, and Step 2(c) gives, for such $m$,

$$x_{k}\,g_{m}(x_{k})=x_{k}^{2},\qquad g_{m}'(x_{k})=1,\qquad\bigl(g_{m}(x_{k})-x_{k}\bigr)^{2}=0 .$$

Hence, pointwise on $X$, the three sequences converge to $x_{k}^{2}$, $1$ and $0$. The functions $x\mapsto x_{k}g_{m}(x_{k})=x_{k}\varphi_{m}(x)$, $x\mapsto g_{m}'(x_{k})=\partial_{k}\varphi_{m}(x)$ and $x\mapsto(\varphi_{m}(x)-x_{k})^{2}$ are Borel, by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel, Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By Step 2(b), $|g_{m}(t)-t|\le|g_{m}(t)|+|t|\le(M+1)|t|$ for $t\in\mathbb{R}$, so for all $x$ and $m$

$$|x_{k}\,g_{m}(x_{k})|\le M\,x_{k}^{2},\qquad|g_{m}'(x_{k})|\le M,\qquad\bigl(g_{m}(x_{k})-x_{k}\bigr)^{2}\le(M+1)^{2}x_{k}^{2}.$$

The dominating functions are integrable: $x\mapsto x_{k}^{2}$ by Step 1 and its multiples by Linearity and Monotonicity of the Lebesgue Integral §integrable; the constant function $M$ belongs to $\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §linear and so is integrable by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable. Claim 3 of Dominated Convergence Theorem therefore yields, as $m\to\infty$,

$$\int_{X}x_{k}\,g_{m}(x_{k})\,\nu(dx)\to m_{k},\qquad\int_{X}g_{m}'(x_{k})\,\nu(dx)\to\int_{X}1\,\nu(dx)=\nu(X)=1,\qquad\int_{X}\bigl(\varphi_{m}(x)-x_{k}\bigr)^{2}\,\nu(dx)\to0,\tag{2}$$

where $\int_{X}1\,\nu(dx)=\nu(X)$ by The Integral of an Indicator Function is the Measure of the Set with $A=X$, and $\nu(X)=1$ as $\nu\in\mathcal{P}(X)$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures); the third limit is the integral of the zero function, which is $0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral with the null set $N=\emptyset$.

Step 6 (convergence of the left-hand side of (1)). In $L^{2}(\nu)$ the difference of the classes of $\varphi_{m}$ and $x_{k}$ is the class of $\varphi_{m}-x_{k}$ by The Lebesgue Space of Power-Integrable Functions §space, and by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and Real Inner Product Space §norm its squared norm is $\int_{X}(\varphi_{m}(x)-x_{k})^{2}\,\nu(dx)$. Using the bilinearity and symmetry of the inner product (Real Inner Product Space §inner-product) and The Cauchy-Schwarz Inequality in a Real Inner Product Space in the real inner product space $L^{2}(\nu)$,

$$D_{m}^{2}\le\lVert\zeta_{k}\rVert_{L^{2}(\nu)}^{2}\int_{X}\bigl(\varphi_{m}(x)-x_{k}\bigr)^{2}\,\nu(dx),\qquad D_{m}=\langle\zeta_{k},\varphi_{m}\rangle_{L^{2}(\nu)}-\langle\zeta_{k},x_{k}\rangle_{L^{2}(\nu)}=\langle\zeta_{k},\varphi_{m}-x_{k}\rangle_{L^{2}(\nu)} .$$

By (2) and the limit law for scalar multiples, the right-hand side tends to $0$. Given $\varepsilon>0$ it is therefore eventually less than $\varepsilon^{2}$, so $D_{m}^{2}<\varepsilon^{2}$ and hence $|D_{m}|<\varepsilon$ eventually. Thus $\langle\zeta_{k},\varphi_{m}\rangle_{L^{2}(\nu)}\to\langle\zeta_{k},x_{k}\rangle_{L^{2}(\nu)}$.

Step 7 (claim 1). By (2) and the limit laws for scalar multiples and differences, the right-hand side of (1) tends to $m_{k}/c_{k}-1$ as $m\to\infty$, while by Step 6 the left-hand side tends to $\langle\zeta_{k},x_{k}\rangle_{L^{2}(\nu)}$. The two sides of (1) form the same sequence, so by uniqueness of limits (claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences)

$$\langle\zeta_{k},x_{k}\rangle_{L^{2}(\nu)}=\frac{1}{c_{k}}\int_{X}x_{k}^{2}\,\nu(dx)-1 .$$

Step 8 (claim 2). Let $b$ be admissible. Since $\nu\in\mathcal{P}^{a}_{\rho}$, Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §field provides $V_{b}(\nu)\in L^{2}(\nu;X^{a})$ whose coordinate along $f_{k}$ is the class of $x\mapsto a_{k}^{1/2}b_{k}x_{k}$, which is $a_{k}^{1/2}b_{k}$ times the class $x_{k}$ of Step 1 by The Lebesgue Space of Power-Integrable Functions §space; and by The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field, applicable since $\nu\in\mathcal{P}_{2}(X)$ has a relative score and finite Fisher information relative to $\gamma_{c}$ with weights $a$, the coordinate of $Z^{a}_{\nu}$ along $f_{k}$ is $a_{k}^{1/2}\zeta_{k}$. The space $L^{2}(\nu;X^{a})$ is that of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis on $(X,\mathcal{B}(X),\nu)$ with $E=X^{a}$ and the orthonormal basis $(f_{k})_{k\in\mathbb{N}}$, by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields; so by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates the series

$$\sum_{k=1}^{\infty}\bigl\langle a_{k}^{1/2}\zeta_{k},\,a_{k}^{1/2}b_{k}x_{k}\bigr\rangle_{L^{2}(\nu)}$$

converges, with sum $\langle Z^{a}_{\nu},V_{b}(\nu)\rangle_{\nu}$. By bilinearity of the inner product (Real Inner Product Space §inner-product), $a_{k}^{1/2}a_{k}^{1/2}=a_{k}$ ($a_{k}^{1/2}$ being the nonnegative square root of $a_{k}$, whose square is $a_{k}$ by Existence and Uniqueness of the Nonnegative Square Root) and claim 1, its $k$-th term equals

$$a_{k}^{1/2}a_{k}^{1/2}\,b_{k}\,\langle\zeta_{k},x_{k}\rangle_{L^{2}(\nu)}=a_{k}\,b_{k}\Bigl(\frac{1}{c_{k}}\int_{X}x_{k}^{2}\,\nu(dx)-1\Bigr).$$

Hence the series of claim 2 is this series term by term; it converges, and its sum is $\langle Z^{a}_{\nu},V_{b}(\nu)\rangle_{\nu}$.