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

TL;DR

This paper introduces a precise, dimension-free simulation method for constrained minimization of information-theoretic distances like Kullback-Leibler and Shannon entropy, applicable across various fields without requiring convexity assumptions.

Contribution

The paper develops a novel pure simulation approach for constrained minimization problems involving information distances, with minimal assumptions and broad applicability.

Findings

Method converges precisely in the limit

Applicable to arbitrary dimensions and constraints

Provides new ways to construct useful divergences

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).