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

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, the space $L^{2}(\Omega;\mathbb{R}^{d})$ with its inner product $\langle\cdot,\cdot\rangle_{L^{2}}$, norm $\lVert\cdot\rVert_{L^{2}}$, metric $d_{L^{2}}$, differences, scalar multiples and constant classes $c_{a}$ is that of that clause; it is a real Hilbert space and is open in itself, so that the differential calculus in force there applies to real-valued functions on it. Write $\mathrm{Sym}$ for the set of bounded symmetric bilinear forms on $L^{2}(\Omega;\mathbb{R}^{d})$, with its norm $\lVert\cdot\rVert$, its order $\preceq$, its sums and scalar multiples, its identity form $I$ and its metric $d_{\mathrm{Sym}}$, all as fixed there. The classes $C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$ and $C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$, the gradient $D\Phi(X)\in L^{2}(\Omega;\mathbb{R}^{d})$ and the Hessian $D^{2}\Phi(X)\in\mathrm{Sym}$ are those defined there.

Let $\gamma=(c_{e_{1}},\dots,c_{e_{d}})$ be the constant tuple, which is orthonormal by that clause, and let $N\in\mathrm{Sym}$ be the tail form it determines. For $b\in\mathrm{Sym}$ let $b^{\flat}$ be the compression of $b$ along $\gamma$, an element of $\mathcal{S}(d)$; that lemma is read throughout with $L^{2}(\Omega;\mathbb{R}^{d})$ in place of the space written $H$ there, with $d$ in place of the dimension written $m$ there, and with $\gamma$ in place of the tuple written $e$ there, and Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian is read with $m=d$. The set $\mathcal{S}(d)$ with its norm $\lVert\cdot\rVert$, its ordering $\preceq$, its metric $d_{\mathcal{S}(d)}$ and its identity matrix $I_{d}$ is that of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii), the standard basis vectors $e_{1},\dots,e_{d}$ of $\mathbb{R}^{d}$ and the initial segment $[d]$ are those of that clause, and the dot product $z\cdot w$ of points of $\mathbb{R}^{d}$ and the matrix-vector product $Az$ are those of Euclidean Space and Real Matrices: Standing Notation (i) and Euclidean Space and Real Matrices: Standing Notation (ii). As in those settings the symbol $\preceq$ serves for the order of $\mathrm{Sym}$ and for that of $\mathcal{S}(d)$, and $\lVert\cdot\rVert$ for the norm of a form, the norm of a matrix and the Euclidean norm alike; the arguments determine which is meant.

Lifted test functions and their translation Hessians $H_{\Phi}(X)\in\mathcal{S}(d)$ are those of that definition. That a quadruple is approximable by test data from above or from below for a real-valued function on a subset of $L^{2}(\Omega;\mathbb{R}^{d})$ is as defined there, read with $L^{2}(\Omega;\mathbb{R}^{d})$ in place of the space written $E$ there; local maxima and local minima relative to a subset are understood in the metric space $(L^{2}(\Omega;\mathbb{R}^{d}),d_{L^{2}})$; and convergence of a sequence in $\mathcal{S}(d)$ is convergence in $(\mathcal{S}(d),d_{\mathcal{S}(d)})$. A subsequence is indexed by a strictly increasing map $\varphi:\mathbb{N}\to\mathbb{N}$. Finally $|s|$ is the absolute value of $s\in\mathbb{R}$, and we write $2=1+1$, $3=2+1$, $6=3+3$ and $\lVert Z\rVert_{L^{2}}^{2}=\lVert Z\rVert_{L^{2}}\lVert Z\rVert_{L^{2}}$. Then the following hold.

1. (Functions of class $C^{2}$ are lifted test functions)(i) Every $\Phi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ is a lifted test function, and

$$H_{\Phi}(Z)=\bigl(D^{2}\Phi(Z)\bigr)^{\flat}\qquad\text{for every }Z\in L^{2}(\Omega;\mathbb{R}^{d}).$$

2. (Approximable data yield lifted test functions)(ii) Let $A\subseteq L^{2}(\Omega;\mathbb{R}^{d})$, let $w:A\to\mathbb{R}$, let $\bar{X}\in A$, let $p\in L^{2}(\Omega;\mathbb{R}^{d})$ and let $\mathbb{B}\in\mathrm{Sym}$. Suppose the quadruple $(\bar{X},w(\bar{X}),p,\mathbb{B})$ is approximable by test data from above for $w$ on $A$. Then for every positive $\varepsilon\in\mathbb{R}$ there are $Y\in A$ and a lifted test function $\Phi$ such that the function $A\to\mathbb{R}$ with value $w(Z)-\Phi(Z)$ at $Z$ has a local maximum at $Y$ relative to $A$ and

$$\lVert Y-\bar{X}\rVert_{L^{2}}<\varepsilon,\quad\bigl|w(Y)-w(\bar{X})\bigr|<\varepsilon,\quad\lVert D\Phi(Y)-p\rVert_{L^{2}}<\varepsilon,\quad\bigl\lVert H_{\Phi}(Y)-\mathbb{B}^{\flat}\bigr\rVert<\varepsilon .$$

If instead that quadruple is approximable by test data from below for $w$ on $A$, the same conclusion holds with a local minimum in place of a local maximum.

3. (A quadratic perturbation of the data)(iii) Let $A$, $w$, $\bar{X}$, $p$ and $\mathbb{B}$ be as in clause 2, let $q\in L^{2}(\Omega;\mathbb{R}^{d})$ and let $\mu\in\mathbb{R}$. Let $g:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ be given by $g(Z)=\mu\lVert Z-q\rVert_{L^{2}}^{2}$, and let $w-g$ and $w+g$ denote the functions from $A$ to $\mathbb{R}$ whose values at $Z\in A$ are $w(Z)-g(Z)$ and $w(Z)+g(Z)$.

(a)(iv) If $\bigl(\bar{X},(w-g)(\bar{X}),p,\mathbb{B}\bigr)$ is approximable by test data from above for $w-g$ on $A$, then $\bigl(\bar{X},w(\bar{X}),p+2\mu(\bar{X}-q),\mathbb{B}+2\mu I\bigr)$ is approximable by test data from above for $w$ on $A$.

(b)(v) If $\bigl(\bar{X},(w+g)(\bar{X}),p,\mathbb{B}\bigr)$ is approximable by test data from below for $w+g$ on $A$, then $\bigl(\bar{X},w(\bar{X}),p-2\mu(\bar{X}-q),\mathbb{B}-2\mu I\bigr)$ is approximable by test data from below for $w$ on $A$.

4. (Compression of doubled second-order data)(vi) Let $\alpha\in\mathbb{R}$ be positive and let $\mathbb{X},\mathbb{Y}\in\mathrm{Sym}$ satisfy $\mathbb{X}\preceq\mathbb{Y}$, $\lVert\mathbb{X}\rVert\le6\alpha$, $\lVert\mathbb{Y}\rVert\le6\alpha$ and

$$-3\alpha\bigl(\lVert Z\rVert_{L^{2}}^{2}+\lVert W\rVert_{L^{2}}^{2}\bigr)\ \le\ \mathbb{X}(Z,Z)-\mathbb{Y}(W,W)\ \le\ 3\alpha\lVert Z-W\rVert_{L^{2}}^{2}\qquad\text{for all }Z,W\in L^{2}(\Omega;\mathbb{R}^{d}).$$

Then the pair $(\mathbb{X}^{\flat},\mathbb{Y}^{\flat})$ is admitted at $\alpha$, and $(\mathbb{X}+t\,N)^{\flat}=\mathbb{X}^{\flat}$ and $(\mathbb{Y}+t\,N)^{\flat}=\mathbb{Y}^{\flat}$ for every $t\in\mathbb{R}$.

5. (Admitted pairs: compactness and closedness)(vii) Let $\alpha\in\mathbb{R}$ be positive.

(a)(viii) Let $(\mathbb{X}_{n})_{n\in\mathbb{N}}$ and $(\mathbb{Y}_{n})_{n\in\mathbb{N}}$ be sequences in $\mathcal{S}(d)$ such that $(\mathbb{X}_{n},\mathbb{Y}_{n})$ is admitted at $\alpha$ for every $n\in\mathbb{N}$. Then there are a strictly increasing $\varphi:\mathbb{N}\to\mathbb{N}$ and $\mathbb{X},\mathbb{Y}\in\mathcal{S}(d)$ such that the sequence whose $k$-th term is $\mathbb{X}_{\varphi(k)}$ converges to $\mathbb{X}$ and the sequence whose $k$-th term is $\mathbb{Y}_{\varphi(k)}$ converges to $\mathbb{Y}$.

(b)(ix) Let $(\mathbb{X}_{n})_{n\in\mathbb{N}}$ and $(\mathbb{Y}_{n})_{n\in\mathbb{N}}$ be sequences in $\mathcal{S}(d)$ converging to $\mathbb{X}\in\mathcal{S}(d)$ and to $\mathbb{Y}\in\mathcal{S}(d)$ respectively, and suppose $(\mathbb{X}_{n},\mathbb{Y}_{n})$ is admitted at $\alpha$ for every $n\in\mathbb{N}$. Then $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$.