Hardness and Approximability of Dimension Reduction on the Probability Simplex

TL;DR

This paper investigates the computational difficulty of reducing the dimension of probability distributions on the simplex, proving NP-hardness and providing an approximation algorithm for the problem.

Contribution

It establishes the NP-hardness of probability simplex dimension reduction and introduces an approximation algorithm for this problem.

Findings

Proves the problem is strongly NP-hard.

Develops an approximation algorithm for dimension reduction.

Highlights the computational challenges in probability distribution simplification.

Abstract

Dimension reduction is a technique used to transform data from a high-dimensional space into a lower-dimensional space, aiming to retain as much of the original information as possible. This approach is crucial in many disciplines like engineering, biology, astronomy, and economics. In this paper, we consider the following dimensionality reduction instance: Given an n-dimensional probability distribution p and an integer m<n, we aim to find the m-dimensional probability distribution q that is the closest to p, using the Kullback-Leibler divergence as the measure of closeness. We prove that the problem is strongly NP-hard, and we present an approximation algorithm for it.