IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 64, NO. 24, DECEMBER 15, 2016 6533 Polar n-Complex and n-Bicomplex Singular Value Decomposition and Principal Component Pursuit Tak-Shing T. Chan, Member, IEEE, and Yi-Hsuan Yang, Member, IEEE Abstract—Informed by recent work on tensor singular value de- PCP with c =1has a high probability of exact recovery, though composition and circulant algebra matrices, this paper presents a c can be tuned if the conditions are not met. new theoretical bridge that unifies the hypercomplex and tensor- Despite its success, one glaring omission from the original based approaches to singular value decomposition and robust prin- cipal component analysis. We begin our work by extending the PCP is the lack of complex (and hypercomplex) formulations. principal component pursuit to Olariu’s polar n-complex num- In numerous signal processing domains, the input phase has a bers as well as their bicomplex counterparts. In doing so, we have significant meaning. For example in parametric spatial audio, derived the polar n-complex and n-bicomplex proximity opera- spectrograms have not only spectral phases but inter-channel tors for both the 1 - and trace-norm regularizers, which can be phases as well. For that reason alone, we have recently extended used by proximal optimization methods such as the alternating di- rection method of multipliers. Experimental results on two sets of the PCP to the complex and the quaternionic cases [7]. However, audio data show that our algebraically informed formulation out- there exists inputs with dimensionality greater than four, such as performs tensor robust principal component analysis. We conclude microphone array data, surveillance video from multiple cam- with the message that an informed definition of the trace norm can eras, or electroencephalogram (EEG) signals, which exceed the bridge the gap between the hypercomplex and tensor-based ap- capability of quaternions. These signals may instead be repre- proaches. Our approach can be seen as a general methodology for generating other principal component pursuit algorithms with sented by n-dimensional hypercomplex numbers, defined as [8] proper algebraic structures. a = a0 + a1 e1 + ···+ an−1 en−1 , (3) Index Terms—Hypercomplex, tensors, singular value decompo- ∈ sition, principal component, pursuit algorithms. where a0 ,...,an−1 R and e1 ...,en−1 are the imaginary units. Products of imaginary units are defined by an ar- bitrary (n − 1) × (n − 1) multiplication table, and multipli- I. INTRODUCTION cation follows the distributive rule [8]. If we impose the multiplication rules HE robust principal component analysis (RPCA) [1] has received a lot of attention lately in many application areas − T ej ei ,i= j, of signal processing [2]–[5]. The ideal form of RPCA decom- ei ej = (4) × − poses the input X ∈ Rl m into a low-rank matrix L and a sparse 1, 0, or 1,i= j, matrix S: and extend the algebra to include all 2n−1 combinations of min rank(L)+λS0 s.t. X = L + S, (1) imaginary units (formally known as multivectors): L,S a = a0 + a1 e1 + a2 e2 + ... where ·0 returns the number of nonzero matrix elements. Ow- ing to the NP-hardness of the above formulation, the principal + a1,2 e1 e2 + a1,3 e1 e3 + ... component pursuit (PCP) [1] has been proposed to solve this + ...+ a − e e ...e − , (5) relaxed problem instead [6]: 1,2,...,n 1 1 2 n 1 then we have a Clifford algebra [9]. For example, the real, min L∗ + λS1 s.t. X = L + S , (2) L,S complex, and quaternion algebras are all Clifford algebras. Yet previously, Alfsmann [10] suggests two families of 2N- where ·∗ is the trace norm (sum of the singular values), · 1 dimensional hypercomplex numbers suitable for signal pro- is the entrywise -norm, and λ can be set to c/ max(l, m) 1 cessing and argued for their superiority over Clifford algebras. where c is a positive parameter [1], [2]. The trace norm and the One family starts from the two-dimensional hyperbolic numbers -norm are the tightest convex relaxations of the rank and the 1 and the other one starts from the four-dimensional tessarines,1 -norm, respectively. Under somewhat general conditions [1], 0 with dimensionality doubling up from there. Although initially attractive, the 2N-dimensional restriction (which also affects Manuscript received August 26, 2015; revised May 26, 2016 and July 16, 2016; accepted September 3, 2016. Date of publication September 21, 2016; Clifford algebras) seems a bit limiting. For instance, if we have date of current version October 19, 2016. The associate editor coordinating the 100 channels to process, we are forced to use 128 dimensions review of this manuscript and approving it for publication was Prof. Masahiro (wasting 28). On the other hand, tensors can have arbitrary di- Yukawa. 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 mensions, but traditionally they do not possess rich algebraic Academia Sinica Career Development Program. structures. Fortunately, recent work on the tensor singular value The authors are with the Research Center for Information Technology In- novation, Academia Sinica, Taipei 11564, Taiwan (e-mail: takshingchan@ 1 2 citi.sinica.edu.tw; [email protected]). Hyperbolic numbers are represented by a0 + a1 j where j =1 and Digital Object Identifier 10.1109/TSP.2016.2612171 a0 ,a1 ∈ R [10]. Tessarines are almost identical except that a0 ,a1 ∈ C [10]. 1053-587X © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 6534 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 64, NO. 24, DECEMBER 15, 2016 decomposition (SVD) [11], which the authors call the t-SVD, rules [16]. The inverse of p is the number p−1 such that pp−1 =1 has begun to impose more structures on tensors [12]–[14]. Fur- [16]. Olariu named it the polar n-complex algebra because it is thermore, a tensor PCP formulation based on t-SVD has also motivated by the polar representation of a complex number [16] been proposed lately [15]. Most relevantly, Braman [12] has where√ a + jb ∈ C is represented geometrically by its modu- suggested to investigate the relationship between t-SVD and lus a2 + b2 and polar angle arctan(b/a). Likewise, the polar Olariu’s [16] n-complex numbers (for arbitrary n). This is ex- n-complex number in (6) can be represented by its modulus actly what we need, yet the actual work is not forthcoming. So | | 2 2 ··· 2 we have decided to begin our investigation with Olariu’s po- p = a0 + a1 + + an−1 (8) lar n-complex numbers. Of special note is Gleich’s work on the circulant algebra [17], which is isomorphic to Olariu’s po- together with n/2 −1 azimuthal angles, n/2 −2 planar lar n-complex numbers. This observation simplifies our current angles, and one polar angle (two if n is even), totaling n − 1 T work significantly. Nevertheless, the existing tensor PCP [15] angles [16]. To calculate these angles, let [A0 ,A1 ,...,An−1 ] T employs an ad hoc tensor nuclear norm, which lacks algebraic be the discrete Fourier transform (DFT) of [a0 ,a1 ,...,an−1 ] , validity. So, in this paper, we remedy this gap by formulating defined by ⎡ ⎤ ⎡ ⎤ the first proper n-dimensional PCP algorithm using the polar A0 a0 n-complex algebra. ⎢ ⎥ ⎢ ⎥ ⎢ A1 ⎥ ⎢ a1 ⎥ Our contributions in this paper are twofold. First, we have ⎢ ⎥ = F ⎢ ⎥ , (9) ⎣ . ⎦ n ⎣ . ⎦ extended PCP to the polar n-complex algebra and the polar . n-bicomplex algebra (defined in Section III), via: 1) properly An−1 an−1 exploiting the circulant isomorphism for the polar n-complex −j2π/n numbers; 2) extending the polar n-complex algebra to a new where ωn = e is a principal nth root of unity and ⎡ ⎤ polar n-bicomplex algebra; and 3) deriving the proximal opera- 11··· 1 tors for both the polar n-complex and n-bicomplex matrices by ⎢ ··· n−1 ⎥ 1 ⎢ 1 ωn ωn ⎥ leveraging the aforementioned isomorphism. Second, we have √ ⎢ ⎥ Fn = ⎣ . .. ⎦ , (10) provided a novel hypercomplex framework for PCP where al- n . n−1 ··· (n−1)(n−1) gebraic structures play a central role. 1 ωn ωn This paper is organized as follows. In Section II, we review which is unitary, i.e., F∗ = F−1 .Fork =1,...,n/2 −1,the polar n-complex matrices and their properties. We extend this to n n azimuthal angles φ can be calculated from [16] the polar n-bicomplex case in Section III. This leads to the polar k −jφk n-complex and n-bicomplex PCP in Section IV. Experiments Ak = |Ak |e , (11) are conducted in Sections V and VI to justify our approach. We ≤ conclude by describing how our work provides a new direction where 0 φk < 2π. Note that we have reversed the sign of the for future work in Section VII. angles as Olariu was a physicist so his DFT is our inverse DFT. Furthermore, for k =2,...,n/2 −1, the planar angles ψk−1 II. THE POLAR n-COMPLEX NUMBERS are defined by [16] | | In this section we introduce polar n-complex matrices and A1 tan ψk−1 = , (12) their isomorphisms. These will be required in Section IV for the |Ak | formulation of polar n-complex PCP. Please note that the value where 0 ≤ ψk ≤ π/2. The polar angle θ+ is defined as [16] of n here does not have to be a power of two. √ 2|A1 | tan θ+ = , (13) A. Background A0 Olariu’s [16] polar n-complex numbers, which we denote where 0 ≤ θ+ ≤ π.