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

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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 ℓ₁- 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.