Proof of The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions

Each result cited is universally quantified over the data in its own statement. For $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, as in The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings. Two facts about means are used throughout, both from The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean: $\lVert m(\mu)-m(\nu)\rVert\le W_{2}(\mu,\nu)$, and $m((\tau_{a})_{\#}\mu)=m(\mu)+a$. Also $W_{2}(\mu,\nu)\le\sqrt{I(\pi)}$ for every $\pi\in\Pi(\mu,\nu)$, by The Quadratic Wasserstein Distance on Euclidean Space §distance and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim 1. Let $\mu\in Q$. By The Map Property of a Set of Probability Measures §map-property the pair $(\mu,\nu_{0})$ is uniquely mapped, so an optimal map $S_{\mu}$ from $\mu$ to $\nu_{0}$ exists, its class does not depend on the choice by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §unique, and $\mathrm{id}-S_{\mu}\in T_{\mu}$. We verify the five properties of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test.

(a) For $\mu,\mu'\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the triangle inequality and the symmetry of $W_{2}$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry) give $W_{2}(\mu,\nu_{0})\le W_{2}(\mu,\mu')+W_{2}(\mu',\nu_{0})$ and the same with $\mu$ and $\mu'$ exchanged, so $|W_{2}(\mu,\nu_{0})-W_{2}(\mu',\nu_{0})|\le W_{2}(\mu,\mu')$ by claim 6 of Properties of the Absolute Value in an Ordered Field. Hence the map $\mu\mapsto W_{2}(\mu,\nu_{0})$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$, the radius $\varepsilon$ serving for the accuracy $\varepsilon$, and $\psi$, its product with itself, is continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space.

(b) By The Squared Wasserstein Distance to a Fixed Measure: a One-Sided Bound Along Couplings, Differentiability at a Uniquely Mapped Source, and the Translation and Centring Identities §differentiable, $\psi$ is differentiable along couplings at $\mu$ with gradient $2(\mathrm{id}-S_{\mu})$, which lies in $T_{\mu}$ because $\mathrm{id}-S_{\mu}$ does and $T_{\mu}$ is a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed. By Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §gradient, $\nabla\psi(\mu)=2(\mathrm{id}-S_{\mu})$.

(c) Let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $Q$ and let $\pi_{n}\in\Pi(\mu_{n},\mu)$ with $I(\pi_{n})\to0$. For each $n$ let $T_{n}$ be an optimal map from $\mu_{n}$ to $\nu_{0}$, which exists as above since $\mu_{n}\in Q$; by (b), $\nabla\psi(\mu_{n})=2(\mathrm{id}-T_{n})$. By Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space these gradients are represented by $x\mapsto2(x-T_{n}(x))$ and $y\mapsto2(y-S_{\mu}(y))$, and for every $z$,

by the vector arithmetic of $\mathbb{R}^{d}$ and claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, the absolute value of the positive number $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field) being $2$ by Absolute Value in an Ordered Field. By claim 1 of Linearity and Monotonicity of the Lebesgue Integral the discrepancy of $\nabla\psi(\mu_{n})$ and $\nabla\psi(\mu)$ along $\pi_{n}$ is $4$ times the discrepancy of $\mathrm{id}-T_{n}$ and $\mathrm{id}-S_{\mu}$ along $\pi_{n}$, which converges to $0$ by Stability of the Optimal Displacement Under Perturbation of the Source Along Couplings of Vanishing Cost §displacement, read with $\nu_{0}$, $\mu$, $S_{\mu}$ in place of its $T$, $(\mu_{n})$, $(T_{n})$ and $(\pi_{n})$; so the former converges to $0$ by claim 3 of Arithmetic of Limits of Real Sequences.

(d) Let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $a\in\mathbb{R}^{d}$. The translation $\tau_{0_{\mathbb{R}^{d}}}$ is the identity map, so $(\tau_{0_{\mathbb{R}^{d}}})_{\#}\nu_{0}=\nu_{0}$ by the definition of the push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, and The Squared Wasserstein Distance to a Fixed Measure: a One-Sided Bound Along Couplings, Differentiability at a Uniquely Mapped Source, and the Translation and Centring Identities §translation, read with $\rho$, $\nu_{0}$, $a$ and $b=0_{\mathbb{R}^{d}}$, gives

Since $\lVert a\rVert^{2}=a\cdot a$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $a\cdot a=a\cdot(I_{d}a)$ by claim 2 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, and $(2I_{d})a=2(I_{d}a)$ by claim 1 of that lemma, the right-hand side is $\tfrac{1}{2}\,a\cdot\bigl((2I_{d})a\bigr)+q\cdot a+c$ with $q=2(m(\rho)-m(\nu_{0}))$ and $c=W_{2}(\rho,\nu_{0})^{2}$, using $2\,a\cdot(m(\rho)-m(\nu_{0}))=q\cdot a$ by claims 1 and 4 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$. The matrix $2I_{d}$ is symmetric, its entries being unchanged when the indices are interchanged. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, read in dimension $d$ with $M=2I_{d}$, the function $a\mapsto\psi((\tau_{a})_{\#}\rho)$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with Hessian matrix $2I_{d}$ at every point; so property (d) holds, and $H_{\psi}(\rho)=2I_{d}$ by Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian.

(e) By (d) the Hessian at the origin of the function attached to $\psi$ at any $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ is $2I_{d}$. For $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the difference $2I_{d}-2I_{d}$ is the zero matrix $0_{d}$, entrywise by Difference of Real Matrices, and $\lVert0_{d}\rVert=0$ by claim 4 of Properties of the Norm of a Symmetric Real Matrix; so for every positive $\varepsilon$ any positive $\theta$, for instance $\theta=\varepsilon$, serves in property (e).

Claim 2. A first-order bound for $\phi$. Let $a\in\mathbb{R}^{d}$ and let $\varepsilon\in\mathbb{R}$ be positive. We show that there is a positive $r\in\mathbb{R}$ with

Let $g:\mathbb{R}^{d}\to\mathbb{R}$ be $g(b)=\phi(b)-D\phi(a)\cdot b$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, read with $M=0_{d}$, $q=-D\phi(a)$ and $c=0$, the map $b\mapsto-D\phi(a)\cdot b$ is of class $C^{2}$ with constant gradient $-D\phi(a)$, so $g$ is of class $C^{2}$ on $\mathbb{R}^{d}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, with $\partial_{i}g(b)=\partial_{i}\phi(b)-\partial_{i}\phi(a)$ by claim 1 of that lemma. Put $\varepsilon_{1}=\varepsilon\,(\sqrt{d}+1)^{-1}$, positive. By Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §gradient, read in dimension $d$ with $V=\mathbb{R}^{d}$, the gradient map of $\phi$ is continuous at $a$, so there is a positive $r$ with $\lVert D\phi(b')-D\phi(a)\rVert<\varepsilon_{1}$ whenever $\lVert b'-a\rVert<r$; for such $b'$ and each $i\in[d]$, $|\partial_{i}g(b')|\le\lVert D\phi(b')-D\phi(a)\rVert<\varepsilon_{1}$ by claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. Let $b$ satisfy $\lVert b-a\rVert<r$. Every point $a+\tau(b-a)$ with $\tau\in[0,1]$ satisfies $\lVert\tau(b-a)\rVert=\tau\lVert b-a\rVert\le\lVert b-a\rVert<r$ by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so claim (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, read with $W=\mathbb{R}^{d}$, $f=g$ and $M_{1}=\varepsilon_{1}$, gives $|g(b)-g(a)|\le\sqrt{d}\,\varepsilon_{1}\lVert b-a\rVert\le\varepsilon\lVert b-a\rVert$. As $g(b)-g(a)=\phi(b)-\phi(a)-D\phi(a)\cdot(b-a)$ by the second identity of claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, this is $(\dagger)$.

We now verify the five properties of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test for $\chi$ on $Q$.

(a) Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\varepsilon\in\mathbb{R}$ be positive. By Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value there is a positive $r$ with $|\phi(b)-\phi(m(\mu))|<\varepsilon$ whenever $\lVert b-m(\mu)\rVert<r$. If $W_{2}(\mu',\mu)<r$, then $\lVert m(\mu')-m(\mu)\rVert<r$, so $|\chi(\mu')-\chi(\mu)|<\varepsilon$. Thus $\chi$ is continuous.

(b) Let $\mu\in Q$, put $a=m(\mu)$, and let $\eta$ be the class of the constant map with value $D\phi(a)$; then $\eta\in T_{\mu}$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields §constants. Let $\varepsilon\in\mathbb{R}$ be positive and let $r$ be as in $(\dagger)$; put $\theta=r$. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$ satisfy $I(\pi)<\theta^{2}$, so that $\sqrt{I(\pi)}<\theta$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §mean, $\mathcal{J}(\eta,\pi)=D\phi(a)\cdot(m(\nu)-m(\mu))$, and $\lVert m(\nu)-m(\mu)\rVert\le W_{2}(\mu,\nu)\le\sqrt{I(\pi)}<r$. So $(\dagger)$ with $b=m(\nu)$ gives

the last step by claim 5 of Elementary Arithmetic in an Ordered Field. Hence $\chi$ is differentiable along couplings at $\mu$ with gradient $\eta$, and $\nabla\chi(\mu)=\eta$ by Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §gradient.

(c) Let $\mu\in Q$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $Q$, and let $\pi_{n}\in\Pi(\mu_{n},\mu)$ with $I(\pi_{n})\to0$. Put $c_{n}=D\phi(m(\mu_{n}))$ and $c=D\phi(m(\mu))$. By (b) the discrepancy of $\nabla\chi(\mu_{n})$ and $\nabla\chi(\mu)$ along $\pi_{n}$ is the integral against the probability measure $\pi_{n}$ of the constant $\lVert c_{n}-c\rVert^{2}$, which is $\lVert c_{n}-c\rVert^{2}$ by The Integral of an Indicator Function is the Measure of the Set and claim 1 of Linearity and Monotonicity of the Lebesgue Integral. Let $\varepsilon\in\mathbb{R}$ be positive and let $\varepsilon_{1}$ be the lesser of $\varepsilon$ and $1$, positive by claim 9 of Elementary Order Arithmetic in an Ordered Field. By Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §gradient there is a positive $r$ with $\lVert D\phi(b)-c\rVert<\varepsilon_{1}$ whenever $\lVert b-m(\mu)\rVert<r$. Since $I(\pi_{n})\to0$ there is $N\in\mathbb{N}$ with $I(\pi_{n})<r^{2}$ for $n\ge N$, and then $\lVert m(\mu_{n})-m(\mu)\rVert\le W_{2}(\mu_{n},\mu)\le\sqrt{I(\pi_{n})}<r$, so $\lVert c_{n}-c\rVert<\varepsilon_{1}$ and $\lVert c_{n}-c\rVert^{2}<\varepsilon_{1}\varepsilon_{1}\le\varepsilon_{1}\le\varepsilon$, by claims 10 and 2 of Elementary Order Arithmetic in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field. So the discrepancies converge to $0$.

(d) Let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For $a\in\mathbb{R}^{d}$, $\chi((\tau_{a})_{\#}\rho)=\phi(m(\rho)+a)=\phi(a+m(\rho))$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §translation, read with $V=\mathbb{R}^{d}$ and $b=m(\rho)$, for which $V-b=\mathbb{R}^{d}$, this function of $a$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with Hessian matrix $D^{2}\phi(a+m(\rho))$ at $a$; at $a=0_{\mathbb{R}^{d}}$ it is $D^{2}\phi(m(\rho))$. So property (d) holds and $H_{\chi}(\rho)=D^{2}\phi(m(\rho))$ by Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian.

(e) Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\varepsilon\in\mathbb{R}$ be positive. By (d) the Hessian at the origin of the function attached to $\chi$ at $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ is $D^{2}\phi(m(\rho))$. By Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §hessian, read in dimension $d$ with $V=\mathbb{R}^{d}$, the Hessian map of $\phi$ is continuous at $m(\mu)$ into $\mathcal{S}(d)$ with the distance $d_{\mathcal{S}(d)}(A,B)=\lVert A-B\rVert$ of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm, so there is a positive $r$ with $\lVert D^{2}\phi(b)-D^{2}\phi(m(\mu))\rVert<\varepsilon$ whenever $\lVert b-m(\mu)\rVert<r$. Put $\theta=r$. If $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ satisfies $W_{2}(\nu,\mu)<\theta$, then $\lVert m(\nu)-m(\mu)\rVert\le W_{2}(\nu,\mu)<r$, and so $\lVert D^{2}\phi(m(\nu))-D^{2}\phi(m(\mu))\rVert<\varepsilon$. So property (e) holds.