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

2023/02/24 by Ambrosio, Luigi, Brena, Camillo, Conti, Sergio · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2302.12554

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.

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• Mixed-Derivative Total Variation

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