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

Abstract

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 ncomplex 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.

Incremental tensor regularized least squares with multiple right-hand sides

“…Definition 2.2. (identity tensor) [6] The identity tensor I ∈ R n×n×n 3 is the tensor with I (1) being the n × n identity matrix, and other frontal slices being zeros.…”

“…Several forms of the regularization operator L are presented in [12,14] and here we focus on L = I, an identity tensor throughout this paper. The operator * denotes the tensor-tensor t-product introduced in the seminal work [6,7], which has been proved to be a useful tool with a large number of applications, such as image processing [6,11,16,18], signal processing [1,9,10], tensor recovery and robust tensor PCA [8,9], data completion and denoising [4,10,19]. Compared to the tensor least squares (t-LS) of the form min…”

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

“…then the T-characteristic polynomial P T (x) has the expression P T (x) := LCM(P D 1 (x), P D 2 (x), · · · , P Dp (x)), (7) where 'LCM' means the least common multiplier, {1, 2, . .…”

“…On the other hand, the tensor T-product introduced by Kilmer [21] has been proved to be of great use in many areas, such as image processing [21,29,35,37], computer vision [12], signal processing [7,24,25], low rank tensor recovery and robust tensor PCA [23,24], data completion and denoising [16,25,38]. An approach of linearization is provided by the T-product to transfer tensor multiplication to matrix multiplication by the discrete Fourier transformation and the theories of block circulant matrices [6,19].…”

Denoising of pre‐stack seismic data using subspace estimation methods

“…For , we define as the number of non‐zero entries in . The hard‐thresholding operator is defined as [53]The shrinkage thresholding operator is defined as [54, 55]…”