Trade-Off Invariance Principle for Minimizers of Regularized Functionals

MCML Authors

Massimo Fornasier

Prof. Dr.

Principal Investigator

Applied Numerical Analysis

Jona Klemenc

→ Group Massimo Fornasier
 Applied Numerical Analysis

Alessandro Scagliotti

→ Group Massimo Fornasier
 Applied Numerical Analysis

Abstract

In this paper, we consider functionals of the form Hα(u)=F(u)+αG(u) with α∈[0,+∞), where u varies in a set U≠∅ (without further structure). We first show that, excluding at most countably many values of α, we have that infH⋆αG=supH⋆αG, where H⋆α:=argminUHα, which is assumed to be non-empty. We further prove a stronger result that concerns the {invariance of the} limiting value of the functional G along minimizing sequences for Hα. This fact in turn implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of α, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent.

inproceedings

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Math4AiMl 2025

3rd Workshop of UMI Group Mathematics for Artificial Intelligence and Machine Learning. Bari, Italy, Jan 29-31, 2025.

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Authors

M. Fornasier • J. Klemenc • A. Scagliotti

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Links

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Research Area

 A2 | Mathematical Foundations