Numerical considerations and a new implementation for ICS

TL;DR

This paper introduces a numerically stable and efficient implementation of Invariant Coordinate Selection (ICS) using pivoted QR factorization, enhancing its robustness and applicability to rank-deficient data.

Contribution

A novel ICS implementation based on pivoted QR factorization that improves numerical stability and handles rank deficiency, unlike traditional joint diagonalization methods.

Findings

Improved numerical stability in ill-conditioned situations.

Effective handling of rank-deficient data sets.

Enhanced performance demonstrated on artificial and real data.

Abstract

Invariant Coordinate Selection (ICS) is a multivariate data transformation and a dimension reduction method that can be useful in many different contexts. It can be used for outlier detection or cluster identification, and can be seen as an independent component or a non-Gaussian component analysis method. The usual implementation of ICS is based on a joint diagonalization of two scatter matrices, and may be numerically unstable in some ill-conditioned situations. We focus on one-step M-scatter matrices and propose a new implementation of ICS based on a pivoted QR factorization of the centered data set. This factorization avoids the direct computation of the scatter matrices and their inverse and brings numerical stability to the algorithm. Furthermore, the row and column pivoting leads to a rank revealing procedure that allows computation of ICS when the scatter matrices are not full rank. Several artificial and real data sets illustrate the interest of using the new implementation compared to the original one.