Numerical Considerations and a New Implementation for Invariant Coordinate Selection

IF 1.9 Q1 MATHEMATICS, APPLIED SIAM journal on mathematics of data science Pub Date : 2022-07-05 DOI:10.1137/22M1498759
A. Archimbaud, Z. Drmač, K. Nordhausen, Una Radojicic, A. Ruiz-Gazen
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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.
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不变坐标选择的数值考虑与新实现
不变坐标选择(ICS)是一种多变量数据转换和降维方法,在许多不同的情况下都很有用。它可以用于异常值检测或聚类识别,也可以看作是一种独立分量或非高斯分量分析方法。ICS的通常实现是基于两个散射矩阵的联合对角化,并且在某些病态情况下可能在数值上不稳定。我们专注于一步M-散射矩阵,并提出了一种基于中心数据集的枢轴QR因子分解的ICS的新实现。这种因子分解避免了散射矩阵及其逆矩阵的直接计算,并为算法带来了数值稳定性。此外,行和列的枢轴转动导致秩揭示过程,该过程允许在散射矩阵不是满秩时计算ICS。几个人工和真实的数据集说明了与原始实现相比使用新实现的兴趣。
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