Lie Group Methods in Blind Signal Processing

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 O ( n ) and S O ( n ) . 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.

Keywords

• geometric optimization
• independent component analysis
• independent subspace analysis
• lie groups
• lie algebra
• toral subalgebra
• sensors