Absolutely one-homogeneous functional

A functional $J$ is absolutely one-homogeneous when $J(cu)=|c|J(u)$ for every scalar $c$. For a convex such functional,

Let $K:U\to V$ be a bounded linear operator from a real Banach space to a real Hilbert space, and let $J$ be a proper lower-semicontinuous convex absolutely one-homogeneous functional on $U$. The normalization and source relation imply $\lambda =J(u_{\lambda })$ by the Euler identity for a convex one-homogeneous functional. For the norm penalty $J(u)=\Vert u\Vert$ on a Hilbert space, its nonzero-point subgradient is $u/\Vert u\Vert$. Hence $K^{\ast }K{u}_{\lambda }=u_{\lambda }/\Vert u_{\lambda }{\Vert }^2$: $v=u_{\lambda }/\Vert u_{\lambda }\Vert$ is a right singular vector with singular value $\sigma =1/\Vert u_{\lambda }\Vert$, and $K{u}_{\lambda }$ is the corresponding left singular vector. Thus the definition recovers the ordinary operator singular-vector relation using the one-homogeneous norm penalty. Uniqueness of the vector from its data requires extra hypotheses on $K$ and $J$.

For an absolutely one-homogeneous functional $J$ that is convex and $p\in \partial J(u)$,The subgradient inequality at zero gives $⟨p,u⟩\ge J(u)$, and at $2u$ gives the reverse inequality. This is an Euler identity that needs no differentiability.

A nonzero vector $f$ is an eigenfunction of an absolutely one-homogeneous convex functional $J$ with eigenvalue $\lambda$ when $\lambda f\in \partial J(f)$. This nonlinear analogue of an eigenvector underlies exact shrinkage formulas for several variational flows and proximal maps.