Lie Group Methods in Blind Signal Processing.

Dariusz Mika,Jerzy Jozwik

doi:10.3390/s20020440

Abstract

This paper deals with the use of Lie group methods to solve optimization problems in blind signal processing (BSP), including Independent Component Analysis (ICA) and Independent Subspace Analysis (ISA). The paper presents the theoretical fundamentals of Lie groups and Lie algebra, the geometry of problems in BSP as well as the basic ideas of optimization techniques based on Lie groups. Optimization algorithms based on the properties of Lie groups are characterized by the fact that during optimization motion, they ensure permanent bonding with a search space. This property is extremely significant in terms of the stability and dynamics of optimization algorithms. The specific geometry of problems such as ICA and ISA along with the search space homogeneity enable the use of optimization techniques based on the properties of the Lie groups and . An interesting idea is that of optimization motion in one-parameter commutative subalgebras and toral subalgebras that ensure low computational complexity and high-speed algorithms.

Highlights

Blind signal processing (BSP) is currently one of the most attractive and fast-growing signal processing areas with solid theoretical foundations and many practical applications
The specific geometry of problems such as Independent Component Analysis (ICA) and independent subspace analysis (ISA) along with the search space homogeneity enable the use of optimization techniques based on the properties of the Lie groups O(n) and SO(n)
This paper primarily focuses on ICA and ISA problems, which does not, limit the applicability of the described methods to other types of problems

Summary

Introduction

Blind signal processing (BSP) is currently one of the most attractive and fast-growing signal processing areas with solid theoretical foundations and many practical applications. The term “blind processing” originates from the basic feature of these processing methods, i.e., the fact that there is no need to use any training data or a priori knowledge to obtain results These methods include, among others, Independent Component Analysis (ICA), independent subspace analysis (ISA), sparse component analysis (SCA), nonnegative matrix factorization (NMF), singular value decomposition (SVD), principal component analysis (PCA) and minor component analysis (MCA) as well as the related eigenproblem and invariant subspace problem. If there is a limitation in the form of matrix orthogonality, one can use an alternative method that ensures “locked” with the hyper-surface of orthogonal matrices during optimization motion This method uses the group structure of a set of orthogonal square matrices which, apart from the properties of a smooth differential manifold, provides the set with the properties of a special structure known as a Lie group.

Schematic

Visual representation of the toral subalgebra

Experimental Results

). Methods

Conclusions