Polar $n$-Complex and $n$-Bicomplex Singular Value Decomposition and Principal Component Pursuit

TL;DR

This paper introduces a unified hypercomplex tensor approach to singular value decomposition and principal component pursuit, extending existing methods to polar n-complex numbers and bicomplex numbers, with promising experimental results on audio data.

Contribution

It develops a novel theoretical framework unifying hypercomplex and tensor methods for SVD and PCA, including new proximity operators for polar n-complex and bicomplex numbers.

Findings

Outperforms tensor robust PCA on audio data

Derives new proximity operators for hypercomplex regularizers

Bridges hypercomplex and tensor-based approaches

Abstract

Informed by recent work on tensor singular value decomposition and circulant algebra matrices, this paper presents a new theoretical bridge that unifies the hypercomplex and tensor-based approaches to singular value decomposition and robust principal component analysis. We begin our work by extending the principal component pursuit to Olariu's polar $n$-complex numbers as well as their bicomplex counterparts. In so doing, we have derived the polar $n$-complex and $n$-bicomplex proximity operators for both the $\ell_1$- and trace-norm regularizers, which can be used by proximal optimization methods such as the alternating direction method of multipliers. Experimental results on two sets of audio data show that our algebraically-informed formulation outperforms tensor robust principal component analysis. We conclude with the message that an informed definition of the trace norm can bridge the gap between the hypercomplex and tensor-based approaches. Our approach can be seen as a general methodology for generating other principal component pursuit algorithms with proper algebraic structures.