Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs

In the setting of The Real Numbers: Standing Notation and Background and Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation, let levels 1 and 2 be as in Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels, with the level-indexed notation of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §notation, and let $(\kappa,p,j)$ be a mode-restriction link from level 1 to level 2, with mode restriction $p$ and mode embedding $j$. For $i\in\{1,2\}$ let $\mathcal{P}_{i}=(\mathcal{D}_{i},\mathcal{D}_{\Sigma,i},\mathcal{E}_{i},\Sigma_{i})$ be a noise-closed noise penalty pair on $\mathcal{P}^{a}_{\rho_{i}}$ at level $i$, and suppose that $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are compatible. For $i\in\{1,2\}$ let $e_{i}\in\mathbb{R}$ be any number with $e_{i}\le\mathcal{E}_{i}(\sigma)$ for every $\sigma\in\mathcal{D}_{i}$; such a number exists by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §bounded-below taken at level $i$. The reference measure $\rho_{i}$ belongs to $\mathcal{P}^{a}_{\rho_{i}}$ 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 §reference, and $W_{a,i}$ is a metric on $\mathcal{P}^{a}_{\rho_{i}}$ 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 §metric, both taken at level $i$; for $\mu\in\mathcal{D}_{1}$ one has $p_{\#}\mu\in\mathcal{D}_{2}\subseteq\mathcal{P}^{a}_{\rho_{2}}$ by Compatible Noise Penalty Pairs at a Fine and a Coarse Level §compatible and Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so that $W_{a,2}(p_{\#}\mu,\nu)$ is a nonnegative real number for every $\nu\in\mathcal{P}^{a}_{\rho_{2}}$. Upper semicontinuity of a real function on $\mathcal{D}_{i}$ is understood relative to $\mathcal{D}_{i}$ in the metric space $(\mathcal{P}^{a}_{\rho_{i}},W_{a,i})$, and convergent series of real numbers are those of that definition. Let $\delta\in\mathbb{R}$ be positive, let $b_{1},b_{2}\in\mathbb{R}$, and let $U:\mathcal{D}_{1}\to\mathbb{R}$ and $V:\mathcal{D}_{2}\to\mathbb{R}$ be upper semicontinuous functions with

$$U(\mu)\le b_{1}-\delta\,\mathcal{E}_{1}(\mu)\quad(\mu\in\mathcal{D}_{1}),\qquad V(\nu)\le b_{2}-\delta\,\mathcal{E}_{2}(\nu)\quad(\nu\in\mathcal{D}_{2}).$$

For positive $\alpha\in\mathbb{R}$ let $\Psi_{\alpha}:\mathcal{D}_{1}\times\mathcal{D}_{2}\to\mathbb{R}$ have the value

$$\Psi_{\alpha}(\mu,\nu)=U(\mu)+V(\nu)-\tfrac{\alpha}{2}\,W_{a,2}(p_{\#}\mu,\nu)^{2}$$

at $(\mu,\nu)$; the set $\mathcal{D}_{1}\times\mathcal{D}_{2}$ is nonempty because $\mathcal{D}_{i}$ contains $\mathcal{D}_{\Sigma,i}$ (Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair), which is nonempty by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty. Then the following hold.

1. (Bounds)¶ For every positive $\alpha\in\mathbb{R}$ and every $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$,

$$\Psi_{\alpha}(\mu,\nu)\le b_{1}+b_{2}-\delta\bigl(\mathcal{E}_{1}(\mu)+\mathcal{E}_{2}(\nu)\bigr)\le b_{1}+b_{2}-\delta(e_{1}+e_{2});$$

so the nonempty set of values of $\Psi_{\alpha}$ is bounded above, and its supremum, written $M(\alpha)$ in this and the remaining clauses, is a real number.

2. (Diagonal)¶ For every positive $\alpha\in\mathbb{R}$ and every $\mu\in\mathcal{D}_{1}$ one has $(\mu,p_{\#}\mu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ and $U(\mu)+V(p_{\#}\mu)\le M(\alpha)$.

3. (Penalty-weighted increase of the data)¶ Let $s\in\mathbb{R}$, and let $U':\mathcal{D}_{1}\to\mathbb{R}$ and $V':\mathcal{D}_{2}\to\mathbb{R}$ satisfy

$$U(\mu)+s\,\mathcal{E}_{1}(\mu)\le U'(\mu)\quad(\mu\in\mathcal{D}_{1}),\qquad V(\nu)+s\,\mathcal{E}_{2}(\nu)\le V'(\nu)\quad(\nu\in\mathcal{D}_{2}).$$

Let $\alpha\in\mathbb{R}$ be positive, suppose that the function $\Psi'_{\alpha}:\mathcal{D}_{1}\times\mathcal{D}_{2}\to\mathbb{R}$ with value $U'(\mu)+V'(\nu)-\tfrac{\alpha}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}$ at $(\mu,\nu)$ is bounded above, with supremum $M'(\alpha)$, and let $\tau\in\mathbb{R}$ be nonnegative and $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ satisfy $M(\alpha)-\tau\le\Psi_{\alpha}(\mu,\nu)$. Then

$$M(\alpha)-\tau+s\bigl(\mathcal{E}_{1}(\mu)+\mathcal{E}_{2}(\nu)\bigr)\le M'(\alpha).$$

4. (Decreasing the doubling strength)¶ Let $\alpha,\alpha',\tau\in\mathbb{R}$ satisfy $0<\alpha'<\alpha$ and $0\le\tau$, and let $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ satisfy $M(\alpha)-\tau\le\Psi_{\alpha}(\mu,\nu)$. Then

$$M(\alpha)-\tau+\tfrac{\alpha-\alpha'}{2}\,W_{a,2}(p_{\#}\mu,\nu)^{2}\le M(\alpha').$$

¶In particular $M(\alpha)\le M(\alpha')$ whenever $0<\alpha'<\alpha$.

5. (Perturbed maximisers)¶ Let $m\in\mathbb{R}$ satisfy $m\le M(\alpha)$ for every positive $\alpha\in\mathbb{R}$; by clause 2, $m=U(\mu_{0})+V(p_{\#}\mu_{0})$ is such a number for every $\mu_{0}\in\mathcal{D}_{1}$. ¶There are $K,B_{1},B_{2}\in\mathbb{R}$ with $0\le B_{1}$ and $0\le B_{2}$, where $K$ depends only on $\delta$, $b_{1}$, $b_{2}$, $e_{1}$, $e_{2}$ and $m$, and $B_{i}$ only on $K$ and $\mathcal{P}_{i}$, and none of them on $\alpha$ or $\tau$, with

$$W_{a,1}(\sigma,\rho_{1})\le B_{1}\quad\text{for every }\sigma\in\mathcal{D}_{1}\text{ with }\mathcal{E}_{1}(\sigma)\le K,\qquad W_{a,2}(\sigma,\rho_{2})\le B_{2}\quad\text{for every }\sigma\in\mathcal{D}_{2}\text{ with }\mathcal{E}_{2}(\sigma)\le K,$$

such that the following holds for all $\alpha,\tau\in\mathbb{R}$ with $0<\alpha$ and $0<\tau<1$. There are $(\hat{\mu},\hat{\nu})\in\mathcal{D}_{1}\times\mathcal{D}_{2}$, a sequence $(\mu_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}_{1}$, a sequence $(\nu_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}_{2}$, and positive real numbers $c_{k}$ ($k\in\mathbb{N}$) whose series converges with $\sum_{k=1}^{\infty}c_{k}\le\tau$, such that: ¶$\mathcal{E}_{1}(\hat{\mu})\le K$, $\mathcal{E}_{2}(\hat{\nu})\le K$, and $\mathcal{E}_{1}(\mu_{k})\le K$ and $\mathcal{E}_{2}(\nu_{k})\le K$ for every $k\in\mathbb{N}$; ¶$M(\alpha)-\tau\le\Psi_{\alpha}(\hat{\mu},\hat{\nu})$; ¶for every $(\mu,\nu)\in\mathcal{P}^{a}_{\rho_{1}}\times\mathcal{P}^{a}_{\rho_{2}}$ the series $\sum_{k=1}^{\infty}c_{k}\bigl(W_{a,1}(\mu,\mu_{k})^{2}+W_{a,2}(\nu,\nu_{k})^{2}\bigr)$ converges; and, writing $\Phi(\mu,\nu)$ for $\Psi_{\alpha}(\mu,\nu)$ minus the sum of that series ($(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$),

$$\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})\qquad\text{for every }(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}\text{ with }(\mu,\nu)\ne(\hat{\mu},\hat{\nu}).$$