The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, so that $\tilde{\mu}_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$ for every $n\in\mathbb{N}$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head. For $n\in\mathbb{N}$, $V_{n}\subseteq L^{2}(\mu)$ is the closed linear subspace of Relative Scores of the Finite-Dimensional Projections on a Hilbert Space: Orthogonal Projections of the Score, and the Weighted Fisher Information as the Limit of Its Cutoffs §subspace and $\Pi_{n}$ the orthogonal projection of $L^{2}(\mu)$ onto it.

1. (The score of the rescaled head)¶ Let $\mu$ have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$, and let $n\in\mathbb{N}$. Then $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$. For $k\in[n]$, if $g_{k}:\mathbb{R}^{n}\to\mathbb{R}$ is a Borel representative of $(\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}})_{k}$, then $\Pi_{n}\zeta_{k}$ is the class of $a_{k}^{-1/2}(g_{k}\circ r_{n})$, a function that is square-integrable with respect to $\mu$ with a class independent of the choice of $g_{k}$, by the change-of-variables formula of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward.

2. (Fisher information of the rescaled head)¶ In the situation of claim 1,

$$\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})=\sum_{k=1}^{n}a_{k}\,\lVert\Pi_{n}\zeta_{k}\rVert_{L^{2}(\mu)}^{2}.$$

3. (Convergence of the head Fisher information)¶ Let $\mu$ have a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. Then $\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\le\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ for every $n\in\mathbb{N}$, and $\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\to\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$ as $n\to\infty$.

4. (The Ornstein-Uhlenbeck functional of the rescaled head)¶ Let $n\in\mathbb{N}$ and let $h\in C_{c}^{\infty}(\mathbb{R}^{n})$ be a test function, with gradient map $\nabla h$. Then $g=h\circ\delta_{n}$ belongs to $C^{2}_{b}(\mathbb{R}^{n})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$, and $h\circ r_{n}=g\circ p_{n}$. With $L^{a}_{\mu}(g)$ the noise Ornstein-Uhlenbeck functional of $\mu$ at $g$ and $\nabla_{a}$ the noise gradient,

$$L^{a}_{\mu}(g)=-\ell^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}(h)\qquad\text{and}\qquad\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}=\lVert\nabla h\rVert_{\tilde{\mu}_{n}},$$

the norms being those of $L^{2}(\mu;X^{a})$ and of $L^{2}(\tilde{\mu}_{n};\mathbb{R}^{n})$.

5. (Converse)¶ Suppose that for every $n\in\mathbb{N}$ the measure $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$, and that there is $C\in\mathbb{R}$ with $\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\le C$ for every $n\in\mathbb{N}$. Then $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})\le C$.