Feature-measure energy and its regularity assumptions

Let $d\ge 1$, let $\mathcal{H}$ be a separable real Hilbert space, and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be the closure of a nonempty convex open set. Let $R:\mathcal{H}\to [0,\mathrm{\infty })$ be differentiable, with differential Lipschitz on bounded sets in the uniform sense that for every bounded $B\subseteq \mathcal{H}$ there is a finite $L_B$ such that $\mathrm{\parallel }d{R}_z-d{R}_{z^{\mathrm{\prime }}}{\mathrm{\parallel }}_{{\mathcal{H}}^{\ast }}\le L_B\mathrm{\parallel }z-z^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathcal{H}}(z,z^{\mathrm{\prime }}\in B),$ and with differential bounded on every sublevel set of $R$: for each $c\ge 0$, $\sup \{\mathrm{\parallel }d{R}_z{\mathrm{\parallel }}_{{\mathcal{H}}^{\ast }}:z\in \mathcal{H},R(z)\le c\}<\mathrm{\infty }\mathrm{.}$ In particular, on each closed convex hull of a bounded feature image $\mathrm{\Phi }(Q_r)$, both a finite Lipschitz constant $L_r$ and a finite norm bound $N_r$ for $dR$ are available. Let $\mathrm{\Phi }:\mathrm{\Omega }\to \mathcal{H}$ be Fréchet differentiable and let $V:\mathrm{\Omega }\to [0,\mathrm{\infty })$ be ${\lambda }_V$-semiconvex, with the convention that $V(u)-\frac{{\lambda }_V}{2}\mathrm{\mid }u{\mathrm{\mid }}^2$ is convex on $\mathrm{\Omega }$. For a semiconvex function $f$ on a convex domain $D$ with coefficient $\lambda$, define ${\mathrm{\partial }}_{\lambda }f(u)=\{\xi \in {\mathbb{R}}^d:f(v)\ge f(u)+\xi \cdot (v-u)+\frac{\lambda }{2}\mathrm{\mid }v-u{\mathrm{\mid }}^2\text{ for every }v\in D\}\mathrm{.}$ For a locally semiconvex function $g$ on $D$, define its local Fréchet subdifferential by $\mathrm{\partial }g(u)=\{\xi \in {\mathbb{R}}^d:{\mathrm{lim\hspace{0.17em}inf}}_{\begin{array}{c}v\to u,v\in D\\ v\mathrm{\ne }u\end{array}}\frac{g(v)-g(u)-\xi \cdot (v-u)}{\mathrm{\mid }v-u\mathrm{\mid }}\ge 0\}\mathrm{.}$ When $g$ is globally $\lambda$-semiconvex, this local subdifferential agrees with the convexified supporting set ${\mathrm{\partial }}_{\lambda }g(u)$. For every $r>0$, require a nonempty closed convex set $Q_r\subseteq \mathrm{\Omega }$, nested increasingly in $r$, such that $\{u\in \mathrm{\Omega }:\mathrm{dist}(u,Q_r)\le r^{\mathrm{\prime }}\}\subseteq Q_{r+r^{\mathrm{\prime }}}(r,r^{\mathrm{\prime }}>0),$ such that $\mathrm{\Phi }$ and $V$ are bounded on $Q_r$, $d\mathrm{\Phi }$ is Lipschitz on $Q_r$, and there are fixed constants $C_1,C_2>0$ with ${\sup }_{u\in Q_r}(\mathrm{\parallel }d{\mathrm{\Phi }}_u\mathrm{\parallel }+\mathrm{\parallel }\mathrm{\partial }V(u)\mathrm{\parallel })\le C_1+C_{2}r\mathrm{.}$ Here $\mathrm{\partial }V(u)={\mathrm{\partial }}_{{\lambda }_V}V(u)$ is the supporting set on $\mathrm{\Omega }$, and in the growth bound $\mathrm{\parallel }\mathrm{\partial }V(u)\mathrm{\parallel }$ means its minimum element norm, equivalently the bound requires a supporting vector of norm at most $C_1+C_{2}r$ at every $u\in Q_r$. No bound is imposed on the domain normal cone. This is a bounded-selection variant of the source assumptions: Assumption 2.1(iii)(c) in Chizat and Bach explicitly uses the maximal element norm, which implies the minimum-norm bound wherever the supporting set is nonempty. The source does not specify that its maximal bound excludes domain normals. The bounded-selection convention here allows relative-domain supporting sets with unbounded normal vectors; results using this variant require their own proofs. For a Borel set $A\subseteq \mathrm{\Omega }$, write ${\mathcal{P}}_2(A)$ for the Borel probability measures on ${\mathbb{R}}^d$ concentrated on $A$ and having finite second moment. For a finite nonnegative Borel measure $\mu$ on ${\mathbb{R}}^d$, set $F(\mu )=R\text{\hspace{0.17em}⁣}({\int }_{\mathrm{\Omega }}\mathrm{\Phi }(u)d\mu (u))+{\int }_{\mathrm{\Omega }}V(u)d\mu (u)$ when $\mu$ is concentrated on $\mathrm{\Omega }$, the Bochner integral exists, and the scalar integral is finite; set $F(\mu )=+\mathrm{\infty }$ otherwise. For $\mu \in {\mathcal{P}}_2(\mathrm{\Omega })$ with finite $F(\mu )$, write $m_{\mu }={\int }_{\mathrm{\Omega }}\mathrm{\Phi }d\mu$ and define its first-variation potential on $\mathrm{\Omega }$ by $F^{\mathrm{\prime }}(\mu )(u)=d{R}_{m_{\mu }}[\mathrm{\Phi }(u)]+V(u)\mathrm{.}$ No convexity of $R$ is required by this definition.

Hilbert space

Bochner-integrable function

Bochner integrability criterion

Chizat and Bach, Assumptions 2.1 and Definitions 2.2/2.4

Ambrosio, Gigli, and Savaré, Chapter 4