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

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, MONTH 2016 1 Polar n -Complex and n -Bicomplex Singular V alue Decomposition and Principal Component Pursuit Tak-Shing T. Chan, Member, IEEE and Yi-Hsuan Y ang, Member, IEEE 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 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. I NTRODUCTION T HE robust principal component analysis (RPCA) [1] has received a lot of attention lately in many application areas of signal processing [2]-[5]. Owing to the NPhardness of the above formulation, the principal component pursuit (PCP) [1] has been proposed to solve this relaxed problem instead [6]: min L, S ‖L ‖ λ‖S ‖ 1 s.t. X L S, (2) where ‖·‖ is the trace norm (sum of the singular values),‖·‖ 1 is the entrywise 1-norm, andλ can be set toc/ max(l,m) where c is a positive parameter [1], [2]. The trace norm and the 1-norm are the tightest convex relaxations of the rank and Manuscript received August 26, 2015; revised May 26, 2016 and July 16, 2016; accepted September 3, 2016. This work was supported by a grant from the Ministry of Science and Technology under the contract MOST102-2221-E-001-004-MY3 and the Academia Sinica Career Development Program. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Masahiro Y ukawa. The authors are with the Research Center for Information Technology Innovation, Academia Sinica, Taipei 11564, Taiwan (email: taksh-ingchan@citi.sinica.edu.tw;