Generalized singular vector ($\Vert K{u}_{\lambda }\Vert =1,\lambda K^{\ast }K{u}_{\lambda }\in \partial J(u_{\lambda })$)

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 $J$.