Proof of 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

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement above.

Claim 1. Let $T$ be an optimal map from $\mu$ to $\nu$, let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\pi\in\Pi(\mu,\rho)$. By Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map one has $T_{\#}\mu=\nu$, so the class of $T$, and with it that of $\mathrm{id}-T$, belongs to $L^{2}(\mu;\mathbb{R}^{d})$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable.

The maps $\mathrm{pr}_{2}$ and $T\circ\mathrm{pr}_{1}$ from $\mathbb{R}^{d+d}$ to $\mathbb{R}^{d}$ are Borel, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. By Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal §pushforward-pair, read with $d+d$ in place of $m$ and $d$ in place of $n$, the measure

belongs to $\Pi\bigl((\mathrm{pr}_{2})_{\#}\pi,\,(T\circ\mathrm{pr}_{1})_{\#}\pi\bigr)$ and has

Here $(\mathrm{pr}_{2})_{\#}\pi=\rho$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling; and $(T\circ\mathrm{pr}_{1})_{\#}\pi=T_{\#}\bigl((\mathrm{pr}_{1})_{\#}\pi\bigr)=T_{\#}\mu=\nu$, the first equality because for $B\in\mathcal{B}(\mathbb{R}^{d})$ the definition of a push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and the identity $(T\circ\mathrm{pr}_{1})^{-1}(B)=(\mathrm{pr}_{1})^{-1}(T^{-1}(B))$ give $(T\circ\mathrm{pr}_{1})_{\#}\pi(B)=\pi\bigl((\mathrm{pr}_{1})^{-1}(T^{-1}(B))\bigr)=\mu(T^{-1}(B))$. So $\gamma\in\Pi(\rho,\nu)$ and therefore

by The Quadratic Wasserstein Distance on Euclidean Space §distance.

It remains to expand $I(\gamma)$. For $z\in\mathbb{R}^{d+d}$ put $v(z)=y-x$ and $p(z)=x-T(x)$, so that $y-T(x)=v(z)+p(z)$; by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and the bilinearity of the dot product, Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$,

Each of the three functions of $z$ on the right is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions; the first and third are nonnegative, and the middle one is $\pi$-integrable with

by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing, $\mathrm{id}-T$ being represented by $x\mapsto x-T(x)$. The first has integral $I(\pi)$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost, a real number by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite; and the third has integral $\int_{\mathbb{R}^{d}}\lVert x-T(x)\rVert^{2}\mu(dx)=\lVert\mathrm{id}-T\rVert_{\mu}^{2}=W_{2}(\mu,\nu)^{2}$, by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward applied to $\mathrm{pr}_{1}$ with $(\mathrm{pr}_{1})_{\#}\pi=\mu$, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields and by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §cost. All three integrals being real, claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives

which together with the inequality above is claim 1.

Claim 2. Let $T$ be an optimal map from $\mu$ to $\nu$; such a map exists by Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §uniquely-mapped. Write $\varphi(\rho)=W_{2}(\rho,\nu)^{2}$ and $\eta=2(\mathrm{id}-T)$, an element of $L^{2}(\mu;\mathbb{R}^{d})$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §linear, $\mathcal{J}(\eta,\pi)=2\,\mathcal{J}(\mathrm{id}-T,\pi)$ for every coupling $\pi$ out of $\mu$. Claim 1 may therefore be written

since $\varphi(\mu)=W_{2}(\mu,\nu)^{2}$. As $I(\pi)=\sqrt{I(\pi)}\cdot\sqrt{I(\pi)}$ by Existence and Uniqueness of the Nonnegative Square Root, the upper estimate of Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable holds with any $\theta\le\varepsilon$: if $I(\pi)<\theta^{2}$ then $\sqrt{I(\pi)}<\theta\le\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, whence $I(\pi)\le\varepsilon\sqrt{I(\pi)}$ by claim 5 of Elementary Arithmetic in an Ordered Field.

It remains to bound the same quantity from below by $-\varepsilon\sqrt{I(\pi)}$ for all $\pi$ of small cost. Suppose not. Then there is a positive $\varepsilon\in\mathbb{R}$ such that for every $j\in\mathbb{N}$ there are $\rho_{j}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi_{j}\in\Pi(\mu,\rho_{j})$ with

where $\iota(j)$ is the image of $j$ in $\mathbb{R}$ under the canonical map of The Canonical Map from the Natural Numbers to a Field, positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field: it suffices to take $\theta=\iota(j)^{-1}$ in the negation of the estimate. In particular $\sqrt{I(\pi_{j})}$ is positive for every $j$, since otherwise the second inequality would read $\varphi(\rho_{j})-\varphi(\mu)-\mathcal{J}(\eta,\pi_{j})<0$ while $\mathcal{J}(\eta,\pi_{j})=0$ and $\varphi(\rho_{j})=\varphi(\mu)$, the coupling then having cost $0$, so that $W_{2}(\mu,\rho_{j})=0$ by The Quadratic Wasserstein Distance on Euclidean Space §distance and hence $\rho_{j}=\mu$ by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §separation, while $\mathcal{J}(\eta,\pi_{j})=0$ by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §bound.

Fix $j$ and let $\gamma_{j}\in\Pi(\rho_{j},\nu)$ be an optimal coupling, which exists by Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment, so that $I(\gamma_{j})=W_{2}(\rho_{j},\nu)^{2}=\varphi(\rho_{j})$ by Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal. By Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued, applied to $\pi_{j}\in\Pi(\mu,\rho_{j})$ and $\gamma_{j}\in\Pi(\rho_{j},\nu)$, there is a gluing $\sigma_{j}\in\mathcal{P}(\mathbb{R}^{3d})$ of them; write $A=\mathrm{q}_{1}$, $B=\mathrm{q}_{2}$ and $C=\mathrm{q}_{3}$ for its coordinate fields, elements of $L^{2}(\sigma_{j};\mathbb{R}^{d})$ by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §fields, and $\theta_{j}=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma_{j}\in\Pi(\mu,\nu)$ for the composite coupling of Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite. That clause and Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §fields give

Expanding $B-C=(A-C)-(A-B)$ in the real inner product space $L^{2}(\sigma_{j};\mathbb{R}^{d})$ by Elementary Identities in a Real Inner Product Space §expansion,

Split the inner product as

by Elementary Identities in a Real Inner Product Space §bilinear; here $T\circ A$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps and lies in $L^{2}(\sigma_{j};\mathbb{R}^{d})$, since $\int\lVert T\circ A\rVert^{2}d\sigma_{j}=\int\lVert T\rVert^{2}d\mu=M_{2}(\nu)<\infty$ by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward with $A_{\#}\sigma_{j}=\mu$ and by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable.

The first term is computed by pushing forward under $(A,B)$, whose push-forward of $\sigma_{j}$ is $\pi_{j}$: the integrand $(A(w)-T(A(w)))\cdot(A(w)-B(w))$ is the value at $(A,B)(w)$ of $z\mapsto(x-T(x))\cdot(x-y)$, so by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing,

the sign by claim 2 of Zero Products and Elementary Identities in a Field and claim 2 of Linearity and Monotonicity of the Lebesgue Integral. The second term is bounded by the Cauchy--Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space:

Substituting the split into the expansion, and using $\mathcal{J}(\eta,\pi_{j})=2\mathcal{J}(\mathrm{id}-T,\pi_{j})$,

Since $\theta_{j}\in\Pi(\mu,\nu)$, The Quadratic Wasserstein Distance on Euclidean Space §distance gives $\varphi(\mu)=W_{2}(\mu,\nu)^{2}\le I(\theta_{j})$, so the first bracket is nonnegative; the last term is nonnegative; and the middle term is at least $-2\varepsilon_{j}\sqrt{I(\pi_{j})}$ by the Cauchy--Schwarz bound and claim 3 of Properties of the Absolute Value in an Ordered Field. Hence

and dividing by the positive $2\sqrt{I(\pi_{j})}$, by claim 7 of Elementary Order Arithmetic in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field,

We contradict this by showing that $\varepsilon_{j}\to0$. First, $\lVert T\circ A-C\rVert_{\sigma_{j}}^{2}=\int_{\mathbb{R}^{d+d}}\lVert T(x)-y\rVert^{2}\,\theta_{j}(dz)$, by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward applied to $(A,C)$, whose push-forward of $\sigma_{j}$ is $\theta_{j}$, together with Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections; this is the integral $\int D_{T}\,d\theta_{j}$ of Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost.

Next, $(I(\theta_{j}))_{j\in\mathbb{N}}$ converges to $W_{2}(\mu,\nu)^{2}$. Indeed $W_{2}(\mu,\nu)^{2}\le I(\theta_{j})$ as above, while Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite gives

Now $\sqrt{I(\pi_{j})}<\iota(j)^{-1}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and $W_{2}(\mu,\rho_{j})\le\sqrt{I(\pi_{j})}<\iota(j)^{-1}$ by The Quadratic Wasserstein Distance on Euclidean Space §distance and the same claim, so the triangle inequality The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, with the symmetry The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry, gives

Both bounds tend to their limits by The Archimedean Property of the Real Numbers, so $(\sqrt{I(\theta_{j})})_{j\in\mathbb{N}}$ is squeezed between $W_{2}(\mu,\nu)$ and $2\iota(j)^{-1}+W_{2}(\mu,\nu)$ and converges to $W_{2}(\mu,\nu)$ by claim 2 of Order Properties of Limits of Real Sequences; hence $(I(\theta_{j}))_{j\in\mathbb{N}}$ converges to $W_{2}(\mu,\nu)^{2}$ by claim 3 of Arithmetic of Limits of Real Sequences applied to the product of that sequence with itself, each term being the square of its square root by Existence and Uniqueness of the Nonnegative Square Root.

The pair $(\mu,\nu)$ being uniquely mapped with optimal map $T$, Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost §stability applied to the sequence $(\theta_{j})_{j\in\mathbb{N}}$ in $\Pi(\mu,\nu)$ now gives that $(\int D_{T}\,d\theta_{j})_{j\in\mathbb{N}}$, that is $(\varepsilon_{j}^{2})_{j\in\mathbb{N}}$, converges to $0$. But $\tfrac{1}{2}\varepsilon<\varepsilon_{j}$ for every $j$, with $\tfrac{1}{2}\varepsilon$ positive by claim 8 of Elementary Order Arithmetic in an Ordered Field and $\varepsilon_{j}$ nonnegative, so claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $(\tfrac{1}{2}\varepsilon)^{2}<\varepsilon_{j}^{2}$ for every $j$, with $(\tfrac{1}{2}\varepsilon)^{2}$ positive by claim 5 of Elementary Order Arithmetic in an Ordered Field. This contradicts the convergence of $(\varepsilon_{j}^{2})_{j\in\mathbb{N}}$ to $0$, by Limit of a Sequence of Real Numbers applied with $(\tfrac{1}{2}\varepsilon)^{2}$ in the role of the tolerance there.

This contradiction proves that for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\theta_{-}$ such that $-\varepsilon\sqrt{I(\pi)}\le\varphi(\rho)-\varphi(\mu)-\mathcal{J}(\eta,\pi)$ whenever $I(\pi)<\theta_{-}^{2}$. Taking $\theta=\min\{\varepsilon,\theta_{-}\}$, the minimum of the two, which is positive by claim 2 of Elementary Properties of the Minimum of Two Elements and at most each of them by claim 1 there, both estimates hold at every $\pi$ with $I(\pi)<\theta^{2}$, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field; so the two-sided bound of claim 6 of Properties of the Absolute Value in an Ordered Field gives the estimate of Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable. This establishes claim 2.

Claim 3. Let $a,b\in\mathbb{R}^{d}$ and write $\mu_{a}=(\tau_{a})_{\#}\mu$ and $\nu_{b}=(\tau_{b})_{\#}\nu$, elements of $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable, and $\kappa=2(a-b)\cdot(m(\mu)-m(\nu))+\lVert a-b\rVert^{2}$.

Let $\Theta:\mathbb{R}^{d+d}\to\mathbb{R}^{d+d}$ be the Borel map $z\mapsto\iota^{d,d}(x+a,y+b)$, Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing. If $\pi\in\Pi(\mu,\nu)$ then $\Theta_{\#}\pi=(\tau_{a}\circ\mathrm{pr}_{1},\tau_{b}\circ\mathrm{pr}_{2})_{\#}\pi$ belongs to $\Pi(\mu_{a},\nu_{b})$ by Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal §pushforward-pair and the computation of the marginals in claim 1; conversely the map built from $-a$ and $-b$ sends $\Pi(\mu_{a},\nu_{b})$ into $\Pi(\mu,\nu)$ and is inverse to it, the two composites being the identity, so $\pi\mapsto\Theta_{\#}\pi$ is a bijection of $\Pi(\mu,\nu)$ onto $\Pi(\mu_{a},\nu_{b})$.

For the costs, $(x+a)-(y+b)=(x-y)+(a-b)$, so by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$,

The middle function is $\pi$-integrable with integral $2(a-b)\cdot(m(\nu)-m(\mu))\cdot(-1)$, more precisely: by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §mean, applied with the constant field of value $a-b$, $\int(a-b)\cdot(y-x)\,\pi(dz)=(a-b)\cdot(m(\nu)-m(\mu))$, so $\int(a-b)\cdot(x-y)\,\pi(dz)=(a-b)\cdot(m(\mu)-m(\nu))$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral and Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$. Integrating the display against $\pi$, using claim 2 of Linearity and Monotonicity of the Lebesgue Integral and $\int\lVert a-b\rVert^{2}d\pi=\lVert a-b\rVert^{2}$ by The Integral of an Indicator Function is the Measure of the Set, and applying the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward to $\Theta$ on the left,

By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment there is an optimal $\pi_{0}\in\Pi(\mu,\nu)$, so $I(\pi_{0})=W_{2}(\mu,\nu)^{2}$ by Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal. The number $W_{2}(\mu,\nu)^{2}+\kappa$ is then a lower bound for $\{I(\pi'):\pi'\in\Pi(\mu_{a},\nu_{b})\}$: every such $\pi'$ is $\Theta_{\#}\pi$ for a unique $\pi\in\Pi(\mu,\nu)$, by the bijection above, and $I(\pi')=I(\pi)+\kappa\ge W_{2}(\mu,\nu)^{2}+\kappa$ by The Quadratic Wasserstein Distance on Euclidean Space §distance and the compatibility of the order with addition, an axiom of Ordered Field. It also belongs to that set, being $I(\Theta_{\#}\pi_{0})$. A lower bound belonging to the set is its greatest lower bound, so

by The Quadratic Wasserstein Distance on Euclidean Space §distance, the squared distance being that greatest lower bound. This is claim 3.

Claim 4. Apply claim 3 with $\bar{\mu}$ and $\bar{\nu}$ in place of $\mu$ and $\nu$ and with $a=m(\mu)$, $b=m(\nu)$. By The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §centring one has $(\tau_{m(\mu)})_{\#}\bar{\mu}=\mu$ and $(\tau_{m(\nu)})_{\#}\bar{\nu}=\nu$, and $m(\bar{\mu})=m(\bar{\nu})=0_{\mathbb{R}^{d}}$. Hence the cross term $2(m(\mu)-m(\nu))\cdot(m(\bar{\mu})-m(\bar{\nu}))$ vanishes, and claim 3 reads

which is claim 4.