On extremal points for some vectorial total variation seminorms

TL;DR

This paper characterizes extremal points of vectorial total variation seminorms, revealing their structure in various cases and implications for sparse solutions and continuum mechanics models.

Contribution

It provides a comprehensive analysis of extremal points for different total variation seminorms, including new examples and structural insights.

Findings

Extremals are piecewise constant with two regions in simple cases.

More complex matrix norms yield extremals with multiple regions.

Radial vector fields are extremal for Frobenius total variation in the plane.

Abstract

We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These are central to the understanding of sparse solutions and sparse optimization algorithms for variational problems posed among such functions. For cases in which either the domain or the target are one dimensional or the sum of the total variations of each component is used, we prove that these extremals consist of piecewise constant functions with two regions. For definitions involving more involved matrix norms and in particular spectral norms, we produce families of examples to show that the resulting set of extremal points is larger and includes piecewise constant functions with more than two regions. We also consider the total deformation induced by the symmetrized gradient, for which minimization with linear constraints appears in problems of determination of limit loads in a number of continuum mechanical models involving plasticity. For this case, we show piecewise infinitesimally rigid functions with two pieces to be extremal under mild assumptions. Finally, as an example which is not piecewise constant, we prove that unit radial vector fields are extremal for the Frobenius total variation in the plane.