Multiplicative Algorithm for Orthgonal Groups and Independent Component Analysis

TL;DR

This paper extends a Newton-like multiplicative algorithm to orthogonal groups, providing a second-order convergent method with stability enhancements, applicable to independent component analysis, demonstrated through numerical simulations.

Contribution

It introduces a second-order convergent multiplicative algorithm on orthogonal groups with a Levenberg-Marquardt variation for ICA applications.

Findings

Explicit expression for individual jumps in the algorithm

The algorithm achieves second-order convergence

Numerical simulations demonstrate remarkable performance

Abstract

The multiplicative Newton-like method developed by the author et al. is extended to the situation where the dynamics is restricted to the orthogonal group. A general framework is constructed without specifying the cost function. Though the restriction to the orthogonal groups makes the problem somewhat complicated, an explicit expression for the amount of individual jumps is obtained. This algorithm is exactly second-order-convergent. The global instability inherent in the Newton method is remedied by a Levenberg-Marquardt-type variation. The method thus constructed can readily be applied to the independent component analysis. Its remarkable performance is illustrated by a numerical simulation.