Linear Inverse Problems with Hessian-Schatten Total Variation

TL;DR

This paper characterizes extremal points of the Hessian-Schatten total variation unit ball, revealing their structure and density properties in two dimensions, with implications for inverse problem solutions.

Contribution

It provides a detailed characterization of extremal points of the HTV unit ball and establishes density of CPWL functions in two dimensions, advancing understanding of regularization in inverse problems.

Findings

CPWL functions are dense in the HTV unit ball for dimension 2

Extremal CPWL functions have Hessians with minimal support

Density of CPWL functions implies extremal point closure in 2D

Abstract

In this paper, we characterize the class of extremal points of the unit ball of the Hessian-Schatten total variation (HTV) functional. The underlying motivation for our work stems from a general representer theorem that characterizes the solution set of regularized linear inverse problems in terms of the extremal points of the regularization ball. Our analysis is mainly based on studying the class of continuous and piecewise linear (CPWL) functions. In particular, we show that in dimension $d=2$, CPWL functions are dense in the unit ball of the HTV functional. Moreover, we prove that a CPWL function is extremal if and only if its Hessian is minimally supported. For the converse, we prove that the density result (which we have only proven for dimension $d = 2$) implies that the closure of the CPWL extreme points contains all extremal points.