Linear Independent Component Analysis over Finite Fields: Algorithms and Bounds

TL;DR

This paper investigates linear ICA over finite fields, establishing fundamental bounds, proposing a greedy algorithm that outperforms existing methods, and analyzing the trade-offs between linear and non-linear solutions.

Contribution

It introduces a fundamental lower bound for linear ICA over finite fields and presents a greedy algorithm that surpasses current methods while analyzing its scalability and complexity.

Findings

The proposed greedy algorithm outperforms existing methods.

The overhead of the algorithm decreases with larger problem scales.

A sub-optimal variant reduces computational complexity significantly.

Abstract

Independent Component Analysis (ICA) is a statistical tool that decomposes an observed random vector into components that are as statistically independent as possible. ICA over finite fields is a special case of ICA, in which both the observations and the decomposed components take values over a finite alphabet. This problem is also known as minimal redundancy representation or factorial coding. In this work we focus on linear methods for ICA over finite fields. We introduce a basic lower bound which provides a fundamental limit to the ability of any linear solution to solve this problem. Based on this bound, we present a greedy algorithm that outperforms all currently known methods. Importantly, we show that the overhead of our suggested algorithm (compared with the lower bound) typically decreases, as the scale of the problem grows. In addition, we provide a sub-optimal variant of our suggested method that significantly reduces the computational complexity at a relatively small cost in performance. Finally, we discuss the universal abilities of linear transformations in decomposing random vectors, compared with existing non-linear solutions.