Complex and Quaternionic Principal Component Pursuit and Its Application to Audio Separation

TL;DR

This paper extends principal component pursuit to complex and quaternionic domains to incorporate phase information, improving audio separation tasks like singing voice separation.

Contribution

It introduces complex and quaternionic proximity operators for regularizers and applies them to enhance audio source separation.

Findings

Phase-aware methods outperform real-valued approaches.

Algorithms successfully separate singing voice from music.

Results on iKala and MSD100 datasets validate the approach.

Abstract

Recently, the principal component pursuit has received increasing attention in signal processing research ranging from source separation to video surveillance. So far, all existing formulations are real-valued and lack the concept of phase, which is inherent in inputs such as complex spectrograms or color images. Thus, in this letter, we extend principal component pursuit to the complex and quaternionic cases to account for the missing phase information. Specifically, we present both complex and quaternionic proximity operators for the $\ell_1$- and trace-norm regularizers. These operators can be used in conjunction with proximal minimization methods such as the inexact augmented Lagrange multiplier algorithm. The new algorithms are then applied to the singing voice separation problem, which aims to separate the singing voice from the instrumental accompaniment. Results on the iKala and MSD100 datasets confirmed the usefulness of phase information in principal component pursuit.