Title: Complex and Quaternionic Principal Component Pursuit and Its Application to Audio Separation,
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.