Proof of Lifted Optimal Maps between Rescaled Heads Converge Strongly to the Noise-Optimal Displacement

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

Throughout, $\mu$, $\nu$, $T$ and the maps $S_{n}$ are those of the statement. 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, $\mu,\nu\in\mathcal{P}_{2}(X)$, and 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 §connected the ordered pair $(\mu,\nu)$ is noise-connected, so $W:=W_{a}(\mu,\nu)$ is a nonnegative real number. Since $T$ is a noise-optimal map from $\mu$ to $\nu$, the transport-cost identity gives $\lVert T-\mathrm{id}\rVert_{\mu}^{2}=W^{2}$, hence $\lVert T-\mathrm{id}\rVert_{\mu}=W$, both numbers being nonnegative; and since $T(x)-x\in X^{a}$ and $|T(x)-x|_{a}^{2}=n_{a}(T(x)-x)$ for every $x\in X$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, the norm formula of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations gives

$$\int_{X}n_{a}\bigl(T(x)-x\bigr)\,\mu(dx)=W^{2}. \qquad (0)$$

For Borel maps $g$ and $h$ that can be composed and a Borel measure $\lambda$ on the domain of $g$ one has $(h\circ g)_{\#}\lambda=h_{\#}(g_{\#}\lambda)$, directly from the formula $B\mapsto\lambda(g^{-1}(h^{-1}(B)))$ for push-forwards in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward; this is used without further comment. Composites of Borel maps are Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space.

Step 1 (the equality in claim 1). Fix $n\in\mathbb{N}$ and let $\eta_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the map $\eta_{n}(u)=S_{n}(u)-u$. Its $k$-th component $u\mapsto(S_{n}(u))_{k}-u_{k}$ is the difference of two Borel real functions, hence Borel, so $\eta_{n}$ is Borel by the componentwise criterion. The pairing $(\mathrm{id},S_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n+n}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and $\mathrm{pr}_{1}\circ(\mathrm{id},S_{n})=\mathrm{id}$, $\mathrm{pr}_{2}\circ(\mathrm{id},S_{n})=S_{n}$. Put $\gamma_{n}=(\mathrm{id},S_{n})_{\#}\tilde{\mu}_{n}$, which by hypothesis is an optimal coupling of $\tilde{\mu}_{n}$ and $\tilde{\nu}_{n}$. In particular

$$(S_{n})_{\#}\tilde{\mu}_{n}=(\mathrm{pr}_{2})_{\#}\gamma_{n}=\tilde{\nu}_{n} \qquad (1)$$

by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling. By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward,

$$I(\gamma_{n})=\int_{\mathbb{R}^{n}}\lVert u-S_{n}(u)\rVert^{2}\,\tilde{\mu}_{n}(du)=\int_{\mathbb{R}^{n}}\lVert S_{n}(u)-u\rVert^{2}\,\tilde{\mu}_{n}(du),$$

and $I(\gamma_{n})=W_{2}(\tilde{\mu}_{n},\tilde{\nu}_{n})^{2}$ by Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal. This is the equality in claim 1; in particular $\int_{\mathbb{R}^{n}}\lVert\eta_{n}\rVert^{2}\,d\tilde{\mu}_{n}=W_{2}(\tilde{\mu}_{n},\tilde{\nu}_{n})^{2}<\infty$.

Step 2 (the inequality in claim 1). Fix $n\in\mathbb{N}$. The map $r_{n}$ is Borel by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head and $T$ is Borel, so $r_{n}\circ T$ is Borel, and the pairing $G_{n}=(r_{n},r_{n}\circ T):X\to\mathbb{R}^{n+n}$ is measurable with respect to $\mathcal{B}(X)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing. Let $\gamma'_{n}=(G_{n})_{\#}\mu$. Then $(\mathrm{pr}_{1})_{\#}\gamma'_{n}=(r_{n})_{\#}\mu=\tilde{\mu}_{n}$ and $(\mathrm{pr}_{2})_{\#}\gamma'_{n}=(r_{n})_{\#}(T_{\#}\mu)=(r_{n})_{\#}\nu=\tilde{\nu}_{n}$, because $T_{\#}\mu=\nu$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map; so $\gamma'_{n}$ is a coupling of $\tilde{\mu}_{n}$ and $\tilde{\nu}_{n}$. By change of variables,

$$I(\gamma'_{n})=\int_{X}\bigl\lVert r_{n}(x)-r_{n}(T(x))\bigr\rVert^{2}\,\mu(dx).$$

Fix $x\in X$ and put $w=T(x)-x$, which lies in $X^{a}$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map. Coordinates being linear, the $k$-th component of $r_{n}(T(x))-r_{n}(x)$ is $a_{k}^{-1/2}w_{k}$, so

$$\bigl\lVert r_{n}(x)-r_{n}(T(x))\bigr\rVert^{2}=\sum_{k=1}^{n}a_{k}^{-1}w_{k}^{2}\le|w|_{a}^{2}=n_{a}\bigl(T(x)-x\bigr),$$

the inequality because $|w|_{a}^{2}$ is the supremum of these partial sums by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §partial-sums, and the last equality by the definition of $n_{a}$ in The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel. Integrating this inequality between nonnegative Borel functions and using (0), $I(\gamma'_{n})\le W^{2}$. By The Quadratic Wasserstein Distance on Euclidean Space §distance, $W_{2}(\tilde{\mu}_{n},\tilde{\nu}_{n})^{2}\le I(\gamma'_{n})\le W^{2}$. Together with Step 1 this proves claim 1.

Step 3 (claim 2). Fix $n\in\mathbb{N}$. By Step 1, $\eta_{n}=S_{n}-\mathrm{id}$ is Borel with $\int_{\mathbb{R}^{n}}\lVert\eta_{n}\rVert^{2}\,d\tilde{\mu}_{n}<\infty$, and $\mu\in\mathcal{P}_{2}(X)$, so Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift applies: the map $D_{n}:=\Lambda_{n}\eta_{n}:X\to X$ is Borel, takes its values in $X^{a}$, is measurable as a map into $X^{a}$, satisfies $|D_{n}(x)|_{a}^{2}=\lVert\eta_{n}(r_{n}(x))\rVert^{2}$ for every $x\in X$, and is square-integrable with respect to $\mu$; its class is the element $\Lambda_{n}(S_{n}-\mathrm{id})$ of $L^{2}(\mu;X^{a})$, which is therefore defined. By the norm formula of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, change of variables along $r_{n}$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward) and claim 1,

$$\lVert\Lambda_{n}(S_{n}-\mathrm{id})\rVert_{\mu}^{2}=\int_{X}\lVert\eta_{n}(r_{n}(x))\rVert^{2}\,\mu(dx)=\int_{\mathbb{R}^{n}}\lVert\eta_{n}\rVert^{2}\,d\tilde{\mu}_{n}=W_{2}(\tilde{\mu}_{n},\tilde{\nu}_{n})^{2}\le W^{2}.$$

Taking nonnegative square roots gives claim 2. Since $D_{n}(x)\in X^{a}$, $n_{a}(D_{n}(x))=|D_{n}(x)|_{a}^{2}$ for every $x$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel, so

$$\int_{X}n_{a}\bigl(D_{n}(x)\bigr)\,\mu(dx)=\lVert D_{n}\rVert_{\mu}^{2}\le W^{2}. \qquad (2)$$

Step 4 (graph couplings). By Step 3 and (2), each $D_{n}$ is a noise displacement for $\mu$ in the sense of the preamble of Strong Convergence of Noise Displacement Fields from Weak Convergence of Their Graph Couplings and an Upper Bound on Their Norms; by that preamble, $\mathrm{id}+D_{n}$ and $(\mathrm{id},\mathrm{id}+D_{n})$ are Borel. Put

$$\lambda_{n}=(\mathrm{id}+D_{n})_{\#}\mu\in\mathcal{P}(X),\qquad\pi_{n}=(\mathrm{id},\mathrm{id}+D_{n})_{\#}\mu\in\mathcal{P}(X\times X).$$

The Borel map $\Sigma_{n}=\mathrm{id}+D_{n}$ satisfies $\Sigma_{n}(x)-x=D_{n}(x)\in X^{a}$ for every $x$ and, by (2), $\int_{X}n_{a}(\Sigma_{n}(x)-x)\,\mu(dx)<\infty$; so by Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement, $\pi_{n}\in\Pi^{a}(\mu,\lambda_{n})$ and

$$I^{a}(\pi_{n})=\int_{X}n_{a}\bigl(D_{n}(x)\bigr)\,\mu(dx)=\lVert D_{n}\rVert_{\mu}^{2}\le W^{2}. \qquad (3)$$

Since $\mu\in\mathcal{P}_{2}(X)$ and $\Pi^{a}(\mu,\lambda_{n})$ is nonempty, $\lambda_{n}\in\mathcal{P}_{2}(X)$ by Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound. Likewise the map $v:X\to X$, $v(x)=T(x)-x$, is Borel (preamble of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs), takes values in $X^{a}$ and satisfies $\int_{X}n_{a}(v(x))\,\mu(dx)<\infty$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, so it is a noise displacement for $\mu$, whose class is $T-\mathrm{id}$. Since $x+v(x)=T(x)$ for every $x$, $(\mathrm{id},\mathrm{id}+v)_{\#}\mu=(\mathrm{id},T)_{\#}\mu=:\pi_{T}$, and $\pi_{T}$ is a noise-optimal coupling of $\mu$ and $\nu$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map.

Step 5 ($\lambda_{n}\to\nu$). Fix $n\in\mathbb{N}$. By Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §shift, applied to the Borel map $\eta_{n}$, for every $x\in X$

$$r_{n}\bigl(x+D_{n}(x)\bigr)=r_{n}(x)+\eta_{n}(r_{n}(x))=S_{n}(r_{n}(x)),\qquad Q_{n}\bigl(x+D_{n}(x)\bigr)=Q_{n}x .$$

The first identity says $r_{n}\circ(\mathrm{id}+D_{n})=S_{n}\circ r_{n}$, so $(r_{n})_{\#}\lambda_{n}=(S_{n})_{\#}\tilde{\mu}_{n}=\tilde{\nu}_{n}$ by (1). Let $A_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$, $A_{n}(u)=(a_{1}^{1/2}u_{1},\dots,a_{n}^{1/2}u_{n})$; each component is a real multiple of a coordinate, hence continuous and Borel, so $A_{n}$ is Borel by the componentwise criterion. Since $a_{k}^{1/2}a_{k}^{-1/2}=1$, the coordinate map $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates equals $A_{n}\circ r_{n}$, so

$$(p_{n})_{\#}\lambda_{n}=(A_{n})_{\#}(r_{n})_{\#}\lambda_{n}=(A_{n})_{\#}\tilde{\nu}_{n}=(A_{n})_{\#}(r_{n})_{\#}\nu=(p_{n})_{\#}\nu .$$

The function $x\mapsto|Q_{n}x|^{2}$ is continuous, Borel and nonnegative, with $|Q_{n}x|^{2}\le|x|^{2}$, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity. By change of variables and the second identity,

$$\int_{X}|Q_{n}x|^{2}\,\lambda_{n}(dx)=\int_{X}\bigl|Q_{n}\bigl(x+D_{n}(x)\bigr)\bigr|^{2}\,\mu(dx)=\int_{X}|Q_{n}x|^{2}\,\mu(dx).$$

By Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $(X_{n})_{n\in\mathbb{N}}$ is an exhausting sequence for $X$ and $P_{n}$ is the orthogonal projection onto $X_{n}$, so by Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections §tail, $Q_{n}x\to0_{X}$ for every $x\in X$, that is $|Q_{n}x|\to0$ and hence $|Q_{n}x|^{2}\to0$. The function $x\mapsto|x|^{2}$ is integrable with respect to $\mu$ because $M_{2}(\mu)<\infty$ (The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space). The dominated convergence theorem, applied on $(X,\mathcal{B}(X),\mu)$ to $f_{n}(x)=|Q_{n}x|^{2}$ with limit $0$ and dominating function $x\mapsto|x|^{2}$, gives $\lim_{n\to\infty}\int_{X}|Q_{n}x|^{2}\,\mu(dx)=0$, hence $\lim_{n\to\infty}\int_{X}|Q_{n}x|^{2}\,\lambda_{n}(dx)=0$. Since $\nu\in\mathcal{P}_{2}(X)$ and every $\lambda_{n}\in\mathcal{P}_{2}(X)$ (Step 4), Measures Whose Heads Agree with a Fixed Measure and Whose Tails Vanish Converge in the Quadratic Wasserstein Distance §convergence gives $\lim_{n\to\infty}W_{2}(\lambda_{n},\nu)=0$; hence $\lambda_{n}\Rightarrow\nu$ by Wasserstein Convergence on a Hilbert Space: Weak Convergence, Integrals of Continuous Functions of Quadratic Growth, Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Compactness §weak, and the sequence $(\lambda_{n})_{n\in\mathbb{N}}$ is tight in $(X,d)$ by Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight §weakly-convergent.

Step 6 (subsequential limits of the graph couplings). We show: for every strictly increasing sequence $(m_{i})_{i\in\mathbb{N}}$ in $\mathbb{N}$ there is a strictly increasing sequence $(i_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ with $\pi_{m_{i_{j}}}\Rightarrow\pi_{T}$ as $j\to\infty$. By Tight Family of Borel Measures on a Metric Space §sequence, tightness of the sequence $(\lambda_{n})_{n\in\mathbb{N}}$ (Step 5) means that the set $\{\lambda_{n}:n\in\mathbb{N}\}$ of its terms is tight in $(X,d)$. The set $\{\lambda_{m_{i}}:i\in\mathbb{N}\}$ is contained in $\{\lambda_{n}:n\in\mathbb{N}\}$, so it is tight in $(X,d)$, every compact set witnessing tightness of the larger set witnessing it for the smaller; and $\{\mu\}$ is tight by Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight §tight. By Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight, the set of all couplings in $\Pi(\mu,\lambda_{m_{i}})$, $i\in\mathbb{N}$, is tight in $X\times X$; since $\pi_{m_{i}}\in\Pi^{a}(\mu,\lambda_{m_{i}})\subseteq\Pi(\mu,\lambda_{m_{i}})$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings, the set of terms of the sequence $(\pi_{m_{i}})_{i\in\mathbb{N}}$ is tight, that is, this sequence is tight in $X\times X$ (Tight Family of Borel Measures on a Metric Space §sequence), a metric space with the distance of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs. By Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence there are a strictly increasing sequence $(i_{j})_{j\in\mathbb{N}}$ and $\pi'\in\mathcal{P}(X\times X)$ with $\pi_{m_{i_{j}}}\Rightarrow\pi'$. Put $\mu_{j}=\mu$ and $\nu_{j}=\lambda_{m_{i_{j}}}$. Then $\mu_{j}\Rightarrow\mu$, the integrals in Weak Convergence of Finite Borel Measures on a Metric Space being constant in $j$; and $\nu_{j}\Rightarrow\nu$, because for every bounded continuous $f:X\to\mathbb{R}$ the sequence $(\int_{X}f\,d\nu_{j})_{j}$ is a subsequence, along the strictly increasing indices $j\mapsto m_{i_{j}}$, of the sequence $(\int_{X}f\,d\lambda_{n})_{n}$, which converges to $\int_{X}f\,d\nu$ by Step 5, and so converges to the same limit. By (3), $0\le I^{a}(\pi_{m_{i_{j}}})\le W^{2}$ for every $j$, so this sequence is bounded and its limit inferior is at most $W^{2}$. By Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §lsc, $\pi'\in\Pi^{a}(\mu,\nu)$ and $I^{a}(\pi')\le\liminf_{j}I^{a}(\pi_{m_{i_{j}}})\le W^{2}$, while $W^{2}\le I^{a}(\pi')$ by The Noise Wasserstein Distance §distance. Hence $I^{a}(\pi')=W^{2}$, and $\pi'$ is noise-optimal. Since $(\mu,\nu)$ is uniquely noise-mapped, there is a noise-optimal map $T^{*}$ from $\mu$ to $\nu$ such that every noise-optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T^{*})_{\#}\mu$. Both $\pi'$ and $\pi_{T}$ are noise-optimal couplings of $\mu$ and $\nu$ (Step 4), so $\pi'=(\mathrm{id},T^{*})_{\#}\mu=\pi_{T}$.

Step 7 (claim 3). Suppose that the real sequence $(\lVert\Lambda_{n}(S_{n}-\mathrm{id})-(T-\mathrm{id})\rVert_{\mu})_{n\in\mathbb{N}}$ does not converge to $0$. Then there is a real $\varepsilon>0$ such that for every $N\in\mathbb{N}$ some $n\ge N$ has $\lVert D_{n}-(T-\mathrm{id})\rVert_{\mu}\ge\varepsilon$; choosing such indices recursively, each larger than the previous one, gives a strictly increasing sequence $(m_{i})_{i\in\mathbb{N}}$ with $\lVert D_{m_{i}}-(T-\mathrm{id})\rVert_{\mu}\ge\varepsilon$ for every $i$. Let $(i_{j})_{j\in\mathbb{N}}$ be given by Step 6, so that $\pi_{m_{i_{j}}}\Rightarrow\pi_{T}$. Apply Strong Convergence of Noise Displacement Fields from Weak Convergence of Their Graph Couplings and an Upper Bound on Their Norms §strong-displacement to the noise displacements $v_{j}=D_{m_{i_{j}}}$ and $v=T-\mathrm{id}$ for $\mu$ (Step 4). Its first hypothesis holds because $(\mathrm{id},\mathrm{id}+v_{j})_{\#}\mu=\pi_{m_{i_{j}}}\Rightarrow\pi_{T}=(\mathrm{id},\mathrm{id}+v)_{\#}\mu$. Its second hypothesis holds with any $J$: for every $j$, $\lVert v_{j}\rVert_{\mu}\le W=\lVert v\rVert_{\mu}$ by claim 2 and the transport-cost identity recalled at the start, so $\lVert v_{j}\rVert_{\mu}\le\lVert v\rVert_{\mu}+\varepsilon'$ for every positive real $\varepsilon'$. Hence $\lim_{j\to\infty}\lVert D_{m_{i_{j}}}-(T-\mathrm{id})\rVert_{\mu}=0$, contradicting $\lVert D_{m_{i_{j}}}-(T-\mathrm{id})\rVert_{\mu}\ge\varepsilon$ for every $j$. Therefore, the class of $D_{n}$ being $\Lambda_{n}(S_{n}-\mathrm{id})$ by Step 3, $\lVert\Lambda_{n}(S_{n}-\mathrm{id})-(T-\mathrm{id})\rVert_{\mu}\to0$ as $n\to\infty$, which is claim 3.