Multiplicative Nonholonomic/Newton -like Algorithm

TL;DR

This paper introduces a novel multiplicative algorithm based on fourth order cumulants for stochastic variables, leveraging Lie group properties to achieve second order convergence without prewhitening.

Contribution

It develops a new multiplicative update algorithm utilizing Lie group structure and fourth order cumulants, with proven second order convergence and no need for prewhitening.

Findings

Achieves second order convergence.

Operates without prewhitening.

Handles invariant cost functions in coset spaces.

Abstract

We construct new algorithms from scratch, which use the fourth order cumulant of stochastic variables for the cost function. The multiplicative updating rule here constructed is natural from the homogeneous nature of the Lie group and has numerous merits for the rigorous treatment of the dynamics. As one consequence, the second order convergence is shown. For the cost function, functions invariant under the componentwise scaling are choosen. By identifying points which can be transformed to each other by the scaling, we assume that the dynamics is in a coset space. In our method, a point can move toward any direction in this coset. Thus, no prewhitening is required.