  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 $\ell_1$- 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.
