A precise bare simulation approach to the minimization of some distances. Foundations

Abstract

In information theory -- as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition -- many flexibilizations of the omnipresent Kullback-Leibler information distance (relative entropy) and of the closely related Shannon entropy have become frequently used tools. To tackle corresponding constrained minimization (respectively maximization) problems by a newly developed dimension-free bare (pure) simulation method, is the main goal of this paper. Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit). As a side effect, we also derive an innovative way of constructing new useful distances/divergences. To illustrate the core of our approach, we present numerous solved cases. The potential for widespread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences and entropies in various different research fields (which may also serve as an interdisciplinary interface).

v3: considerably shortened and restructured version of v1/v2; 64 pages + 7 pages supplement. This work is accepted by the journal "IEEE Transactions on Information Theory", and is available in early-access form at https://ieeexplore.ieee.org/document/9925151

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• Some smooth divergences for ${\ell }_1-$approximations