Proof of A Bound on the Noise Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Weighted Fisher Information

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

Throughout, the Euclidean items cited for measures on $\mathbb{R}^{n}$ are read with $n$ in place of the dimension $d$, as fixed in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §euclidean, and $\tilde{\mu}_{n}=(r_{n})_{\#}\mu$ and $\delta_{n}$ are the rescaled head and the rescaling map, those of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads.

Step 1 (the bound for the rescaled heads). Fix $n\in\mathbb{N}$. Since $\mu\in\mathcal{P}_{2}(X)$, the measure $\tilde{\mu}_{n}$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{n})$ by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head, so its Ornstein-Uhlenbeck functional $\ell=\ell^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}$ of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §functional is defined. Let $h\in C_{c}^{\infty}(\mathbb{R}^{n})$ be a test function. By 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 §functional, the function $g=h\circ\delta_{n}$ belongs to $C^{2}_{b}(\mathbb{R}^{n})$, and $L^{a}_{\mu}(g)=-\ell(h)$ and $\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}=\lVert\nabla h\rVert_{\tilde{\mu}_{n}}$. The hypothesis of the lemma, applied to this $n$ and this $g$, together with $|-\ell(h)|=|\ell(h)|$ (Properties of the Absolute Value in an Ordered Field §symmetry), gives

$$|\ell(h)|=\bigl|L^{a}_{\mu}(g)\bigr|\le R\,\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}=R\,\lVert\nabla h\rVert_{\tilde{\mu}_{n}} .$$

As $h$ was arbitrary and $R\ge0$, the measure $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$ by Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §finite, with the constant $R$.

Step 2 (the head Fisher information is at most $R^{2}$). Keep $n$ fixed and let $\zeta=\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}\in T_{\tilde{\mu}_{n}}$ be the relative score of $\tilde{\mu}_{n}$ given by Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §score, the unique element of $T_{\tilde{\mu}_{n}}$ with $\langle\zeta,\nabla h\rangle_{\tilde{\mu}_{n}}=-\ell(h)$ for every $h\in C_{c}^{\infty}(\mathbb{R}^{n})$. The map $-\ell$ is linear on $C_{c}^{\infty}(\mathbb{R}^{n})$, because $\ell$ is linear by Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §functional, and by Step 1 and Properties of the Absolute Value in an Ordered Field §symmetry it satisfies $|-\ell(h)|\le R\,\lVert\nabla h\rVert_{\tilde{\mu}_{n}}$ for every test function $h$, with $R\ge0$. Hence Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §representation, applied at the measure $\tilde{\mu}_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$ to the functional $-\ell$ and the constant $R$, yields $\xi\in T_{\tilde{\mu}_{n}}$ with $\langle\xi,\nabla h\rangle_{\tilde{\mu}_{n}}=-\ell(h)$ for every test function $h$ and $\lVert\xi\rVert_{\tilde{\mu}_{n}}\le R$. By the uniqueness in Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §score, $\xi=\zeta$, so $\lVert\zeta\rVert_{\tilde{\mu}_{n}}\le R$. Both numbers are nonnegative, so the weak form (claim 2) of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert\zeta\rVert_{\tilde{\mu}_{n}}^{2}\le R^{2}$, that is,

$$\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})=\lVert\zeta\rVert_{\tilde{\mu}_{n}}^{2}\le R^{2}$$

by Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information §information.

Step 3 (conclusion). Since $n\in\mathbb{N}$ was arbitrary, for every $n\in\mathbb{N}$ the measure $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$ and $\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\le R^{2}$. Applying 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 §converse to $\mu\in\mathcal{P}_{2}(X)$ with $C=R^{2}$, the measure $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})\le R^{2}$.