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 $d=2$.
Functions with Bounded Hessian–Schatten Variation: Density, Variational, and Extremality Properties
Ambrosio, Luigi
;Brena, Camillo;Conti, Sergio
2023
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 $d=2$.File | Dimensione | Formato | |
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