# Functions with bounded Hessian-Schatten variation: density, variational and extremality properties

## TL;DR

This paper investigates functions with bounded p-Hessian-Schatten total variation, establishing density, extremality, and existence results relevant to inverse problems and machine learning, with implications for approximation and functional minimization.

## Contribution

It proves a density result for CPWL functions, shows not all extremal functions are CPWL, and establishes existence of minimizers in two dimensions.

## Key findings

- CPWL functions are dense relative to p-Hessian-Schatten total variation
- Not all extremal functions are CPWL
- Existence of minimizers in dimension two

## Abstract

In this paper we analyze in detail a few questions related to the theory of functions with bounded $p$-Hessian-Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the $p$-Hessian-Schatten total variation, of continuous piecewise linear (CPWL) functions in any space dimension $d$, using a construction based on a mesh whose local orientation is adapted to the function to be approximated. We show that not all extremal functions with respect to the $p$-Hessian-Schatten total variation are CPWL. Finally, we prove existence of minimizers of certain relevant functionals involving the $p$-Hessian-Schatten total variation in the critical dimension $d=2$.