Proof of Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple

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

Preliminaries. By The Constant Tuple on $L^{2}(\Omega;\mathbb{R}^{d})$: Coordinates, Projection, Tail Form, and Translation-Closed Preimages §tuple the tuple $\gamma$ is orthonormal, so Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions and Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple apply to it as described in the statement. By The Constant Tuple on $L^{2}(\Omega;\mathbb{R}^{d})$: Coordinates, Projection, Tail Form, and Translation-Closed Preimages §coordinates the map $\Lambda^{\sharp}$ determined by $\gamma$ satisfies $\Lambda^{\sharp}a=c_{a}$ for every $a\in\mathbb{R}^{d}$. By Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian §constants we have, for all $a,b\in\mathbb{R}^{d}$ and every $t\in\mathbb{R}$,

and for each $Z\in L^{2}(\Omega;\mathbb{R}^{d})$ the map $J_{Z}:\mathbb{R}^{d}\to L^{2}(\Omega;\mathbb{R}^{d})$ with $J_{Z}(a)=Z+c_{a}$ is continuous and satisfies $J_{Z}(0_{\mathbb{R}^{d}})=Z$. Taking $t=-1$ in the second identity and then using the first, $c_{z}-c_{w}=c_{z}+c_{-w}=c_{z-w}$ for all $z,w\in\mathbb{R}^{d}$, the identity $z-w=z+(-1)w$ in $\mathbb{R}^{d}$ being that of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space; hence $\lVert c_{z}-c_{w}\rVert_{L^{2}}=\lVert z-w\rVert$. Finally $\lVert e_{i}\rVert=1$ for $i\in[d]$ by Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis, so $\lVert c_{e_{i}}\rVert_{L^{2}}=1$.

Proof of clause 1. Let $\Phi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$. By The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2 the function $\Phi$ is of class $C^{1}$, has a second derivative at every point, and its Hessian map is continuous into $(\mathrm{Sym},d_{\mathrm{Sym}})$. The first of these is property (a) of Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians §test.

Fix $Z\in L^{2}(\Omega;\mathbb{R}^{d})$ and write $\phi=\phi_{Z}$ for the function $\phi(a)=\Phi(Z+c_{a})$ on $\mathbb{R}^{d}$. By Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian §derivative the function $\phi$ is of class $C^{1}$ on $\mathbb{R}^{d}$ and

The second partial derivatives. Fix $i,j\in[d]$ and $a\in\mathbb{R}^{d}$, and write $x=Z+c_{a}$ and $b=D^{2}\Phi(x)\in\mathrm{Sym}$. Let $\varepsilon\in\mathbb{R}$ be positive. Since $L^{2}(\Omega;\mathbb{R}^{d})$ is open in itself and $\Phi$ is differentiable at every one of its points, The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §expansion provides a positive $\rho\in\mathbb{R}$ such that all $W,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\lVert W\rVert_{L^{2}}<\rho$ satisfy

Let $s\in\mathbb{R}$ satisfy $0<|s|<\rho$ and put $W=c_{s e_{j}}$ and $Y=c_{e_{i}}$. Then $\lVert W\rVert_{L^{2}}=\lVert s e_{j}\rVert=|s|\,\lVert e_{j}\rVert=|s|<\rho$ by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and $\lVert Y\rVert_{L^{2}}=1$. Moreover $x+W=Z+c_{a}+c_{se_{j}}=Z+c_{a+se_{j}}$, so that by the formula for $\partial_{i}\phi$ displayed above and by Elementary Identities in a Real Inner Product Space §bilinear,

Also $W=c_{se_{j}}=s\,c_{e_{j}}$, so $b(W,Y)=s\,b(c_{e_{j}},c_{e_{i}})$ by the homogeneity of $b$ in its first argument, Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §form. The displayed estimate therefore reads

and dividing by the positive number $|s|$ (claims 4 and 5 of Elementary Arithmetic in an Ordered Field) and using claim 4 of Properties of the Absolute Value in an Ordered Field,

Here $\varepsilon$ was an arbitrary positive real number; applying what precedes to $\tfrac{\varepsilon}{2}$, which is positive and satisfies $\tfrac{\varepsilon}{2}<\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, we obtain for every positive $\varepsilon$ a positive $\rho$ such that the left-hand side above is strictly less than $\varepsilon$ whenever $0<|s|<\rho$. That is the statement that the limit defining the partial derivative of $\partial_{i}\phi$ with respect to the $j$th variable at $a$ exists and equals $b(c_{e_{j}},c_{e_{i}})$. In the notation of clause 4 of C^k Maps on a Euclidean Open Set,

by Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple §compression, the $j$th and $i$th components of $\gamma$ being $c_{e_{j}}$ and $c_{e_{i}}$.

Continuity. The map $a\mapsto Z+c_{a}$ is continuous, being $J_{Z}$; the Hessian map $D^{2}\Phi$ is continuous into $(\mathrm{Sym},d_{\mathrm{Sym}})$; and the map $\mathrm{Sym}\to\mathbb{R}$ with value $c(c_{e_{j}},c_{e_{i}})$ at $c$ is continuous, since $|c(c_{e_{j}},c_{e_{i}})-c'(c_{e_{j}},c_{e_{i}})|\le\lVert c-c'\rVert$ for $c,c'\in\mathrm{Sym}$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §continuity together with $\lVert c_{e_{j}}\rVert_{L^{2}}=\lVert c_{e_{i}}\rVert_{L^{2}}=1$. By claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, applied twice, the composite $a\mapsto\partial_{j}\partial_{i}\phi(a)$ is continuous at every point of $\mathbb{R}^{d}$.

Conclusion of clause 1. Each $\partial_{i}\phi$ is continuous at every point of $\mathbb{R}^{d}$, because $\phi$ is of class $C^{1}$ (clause 1 of C^k Maps on a Euclidean Open Set), and by the two paragraphs above each of its partial derivatives exists at every point and is continuous there. Hence each $\partial_{i}\phi$ is of class $C^{1}$ on $\mathbb{R}^{d}$, and $\phi$ is of class $C^{2}$ on $\mathbb{R}^{d}$ by clauses 2 and 3 of C^k Maps on a Euclidean Open Set. As $Z$ was arbitrary, $\Phi$ is twice continuously differentiable along translations at every point, that is twice continuously differentiable along translations, which is property (b) of Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians §test. So $\Phi$ is a lifted test function.

Finally, by Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians §hessian and Hessian Matrix of a C^2 Function the entry of $H_{\Phi}(Z)=D^{2}\phi_{Z}(0_{\mathbb{R}^{d}})$ in row $i$ and column $j$ is $\partial_{i}\partial_{j}\phi(0_{\mathbb{R}^{d}})$. Applying the computation above with the roles of $i$ and $j$ exchanged and with $a=0_{\mathbb{R}^{d}}$, so that $Z+c_{0_{\mathbb{R}^{d}}}=Z$ by Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian §constants, this entry equals $\bigl((D^{2}\Phi(Z))^{\flat}\bigr)_{ij}$. Two real $d\times d$ matrices with the same entries are equal by Real Matrix and the Set of Real Matrices, so $H_{\Phi}(Z)=(D^{2}\Phi(Z))^{\flat}$.

Proof of clause 2. Assume the quadruple is approximable by test data from above and let $\varepsilon\in\mathbb{R}$ be positive. By Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §above there are $Y\in A$ and $\varphi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ such that $w-\varphi$ has a local maximum at $Y$ relative to $A$ and

By clause 1 the function $\Phi=\varphi$ is a lifted test function with $H_{\varphi}(Y)=(D^{2}\varphi(Y))^{\flat}$. By Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple §algebra, applied to the sum of $D^{2}\varphi(Y)-\mathbb{B}$ and $\mathbb{B}$ and to the scalar multiple of $\mathbb{B}$ by $-1$,

the differences of matrices and of forms being those of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices and Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §identity. The four displayed inequalities of the clause therefore hold with $\Phi=\varphi$. The case from below is the same argument with Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §below and a local minimum in place of a local maximum.

Proof of clause 3. We prove (a); part (b) is the same argument with $-g$ in place of $g$, with Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §below in place of Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §above, and with local minima in place of local maxima.

By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §quadratic, read with $2\mu$ in place of the number written $\alpha$ there and with $q$ in place of $y_{0}$, the function $g$, whose value at $Z$ is $\tfrac{2\mu}{2}\lVert Z-q\rVert_{L^{2}}^{2}$, belongs to $C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ with $Dg(Z)=2\mu(Z-q)$ and $D^{2}g(Z)=2\mu I$ for every $Z$.

Let $\varepsilon\in\mathbb{R}$ be positive and choose a positive $\varepsilon'\in\mathbb{R}$ with

which is possible because each of the four right-hand sides is positive and each left-hand side is a nonnegative multiple of $\varepsilon'$ (claim 9 of Elementary Order Arithmetic in an Ordered Field selects the least of four positive candidates). By Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §above applied to $w-g$ with $\varepsilon'$ there are $Y\in A$ and $\varphi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ such that $(w-g)-\varphi$ has a local maximum at $Y$ relative to $A$ and

Put $\psi=\varphi+g$. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum, $\psi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ with $D\psi(Y)=D\varphi(Y)+2\mu(Y-q)$ and $D^{2}\psi(Y)=D^{2}\varphi(Y)+2\mu I$. The value of $w-\psi$ at $Z\in A$ is $(w(Z)-g(Z))-\varphi(Z)$, that is the value of $(w-g)-\varphi$ at $Z$; hence $w-\psi$ has a local maximum at $Y$ relative to $A$.

It remains to bound the four quantities. First $\lVert Y-\bar{X}\rVert_{L^{2}}<\varepsilon'\le\varepsilon$. Secondly, write $s=\lVert Y-q\rVert_{L^{2}}$ and $t=\lVert\bar{X}-q\rVert_{L^{2}}$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §reverse-triangle, $|s-t|\le\lVert Y-\bar{X}\rVert_{L^{2}}<\varepsilon'$, so $s\le t+\varepsilon'$ and $s+t\le2t+\varepsilon'\le2t+1$ by claim 3 of Properties of the Absolute Value in an Ordered Field and $\varepsilon'\le1$; therefore, by claim 4 of Properties of the Absolute Value in an Ordered Field and the identity $s^{2}-t^{2}=(s-t)(s+t)$ of claim 4 of Zero Products and Elementary Identities in a Field,

the factors $|s-t|$ and $s+t$ being nonnegative (claim 5 of Elementary Arithmetic in an Ordered Field). Since $w(Z)=(w-g)(Z)+g(Z)$ for $Z\in A$, claim 5 of Properties of the Absolute Value in an Ordered Field gives

Thirdly, $D\psi(Y)-\bigl(p+2\mu(\bar{X}-q)\bigr)=\bigl(D\varphi(Y)-p\bigr)+2\mu\,(Y-\bar{X})$, so by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and Elementary Identities in a Real Inner Product Space §homogeneity,

Fourthly, $D^{2}\psi(Y)-(\mathbb{B}+2\mu I)=D^{2}\varphi(Y)-\mathbb{B}$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §identity, so its norm is less than $\varepsilon'\le\varepsilon$. Finally the value recorded by the quadruple is correct, since $(w-g)(\bar{X})+g(\bar{X})=w(\bar{X})$. As $\varepsilon$ was an arbitrary positive real number, $\bigl(\bar{X},w(\bar{X}),p+2\mu(\bar{X}-q),\mathbb{B}+2\mu I\bigr)$ is approximable by test data from above for $w$ on $A$.

Proof of clause 4. Let $z,w\in\mathbb{R}^{d}$. By Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple §compression and $\Lambda^{\sharp}a=c_{a}$,

Applying the hypothesis with $Z=c_{z}$ and $W=c_{w}$ and using $\lVert c_{z}\rVert_{L^{2}}=\lVert z\rVert$, $\lVert c_{w}\rVert_{L^{2}}=\lVert w\rVert$ and $\lVert c_{z}-c_{w}\rVert_{L^{2}}=\lVert z-w\rVert$ from the preliminaries,

By Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple §algebra the compression is order-preserving, so $\mathbb{X}\preceq\mathbb{Y}$ gives $\mathbb{X}^{\flat}\preceq\mathbb{Y}^{\flat}$, and it does not increase the norm, so $\lVert\mathbb{X}^{\flat}\rVert\le\lVert\mathbb{X}\rVert\le6\alpha$ and $\lVert\mathbb{Y}^{\flat}\rVert\le6\alpha$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. These are exactly the four requirements of The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space §admitted, so $(\mathbb{X}^{\flat},\mathbb{Y}^{\flat})$ is admitted at $\alpha$. The two identities $(\mathbb{X}+tN)^{\flat}=\mathbb{X}^{\flat}$ and $(\mathbb{Y}+tN)^{\flat}=\mathbb{Y}^{\flat}$ are Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple §tail.

Proof of clause 5. Throughout, $\lVert\mathbb{X}_{n}\rVert\le6\alpha$ and $\lVert\mathbb{Y}_{n}\rVert\le6\alpha$ for every $n$, by The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space §admitted.

(a) By Limits and Bounded Sequences of Symmetric Real Matrices §compactness, read with $d$ in place of the dimension written $n$ there and with $R=6\alpha$, there are a strictly increasing $\varphi_{1}:\mathbb{N}\to\mathbb{N}$ and $\mathbb{X}\in\mathcal{S}(d)$ such that the sequence whose $k$-th term is $\mathbb{X}_{\varphi_{1}(k)}$ converges to $\mathbb{X}$. The sequence whose $k$-th term is $\mathbb{Y}_{\varphi_{1}(k)}$ lies in $\mathcal{S}(d)$ and is bounded in norm by $6\alpha$, so the same clause provides a strictly increasing $\varphi_{2}:\mathbb{N}\to\mathbb{N}$ and $\mathbb{Y}\in\mathcal{S}(d)$ such that the sequence whose $k$-th term is $\mathbb{Y}_{\varphi_{1}(\varphi_{2}(k))}$ converges to $\mathbb{Y}$. Put $\varphi=\varphi_{1}\circ\varphi_{2}$, which is strictly increasing and indexes a subsequence of the original sequences by A Subsequence of a Subsequence is a Subsequence; the sequence whose $k$-th term is $\mathbb{X}_{\varphi(k)}$ is a subsequence of a sequence converging to $\mathbb{X}$ and therefore converges to $\mathbb{X}$ by A Subsequence of a Convergent Sequence Has the Same Limit.

(b) Let $z,w\in\mathbb{R}^{d}$. By Limits and Bounded Sequences of Symmetric Real Matrices §quadratic-form the sequences whose $n$-th terms are $z\cdot(\mathbb{X}_{n}z)$ and $w\cdot(\mathbb{Y}_{n}w)$ converge to $z\cdot(\mathbb{X}z)$ and to $w\cdot(\mathbb{Y}w)$, so by Arithmetic of Limits of Real Sequences the sequence of their differences converges to $z\cdot(\mathbb{X}z)-w\cdot(\mathbb{Y}w)$. Every term of that sequence lies between $-3\alpha(\lVert z\rVert^{2}+\lVert w\rVert^{2})$ and $3\alpha\lVert z-w\rVert^{2}$, so by Order Properties of Limits of Real Sequences, applied to the constant sequences with these two values, so does its limit. Next, $\mathbb{X}_{n}\preceq\mathbb{Y}_{n}$ for every $n$ gives $\mathbb{X}\preceq\mathbb{Y}$ by Limits and Bounded Sequences of Symmetric Real Matrices §closed. Finally let $\xi\in\mathbb{R}^{d}$ satisfy $\lVert\xi\rVert\le1$. By claim 2 of Properties of the Norm of a Symmetric Real Matrix and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field,

The sequence whose $n$-th term is $\xi\cdot(\mathbb{X}_{n}\xi)$ converges to $\xi\cdot(\mathbb{X}\xi)$, and the sequence of absolute values converges to $|\xi\cdot(\mathbb{X}\xi)|$ because $\bigl||r|-|r'|\bigr|\le|r-r'|$ for real $r,r'$ by claim 7 of Properties of the Absolute Value in an Ordered Field; hence $|\xi\cdot(\mathbb{X}\xi)|\le6\alpha$ by Order Properties of Limits of Real Sequences. Thus $6\alpha$ is an upper bound of the set whose supremum is $\lVert\mathbb{X}\rVert$ in Norm of a Symmetric Real Matrix, so $\lVert\mathbb{X}\rVert\le6\alpha$; the same argument gives $\lVert\mathbb{Y}\rVert\le6\alpha$. The four requirements of The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space §admitted hold, so $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$.