NPSA: Nonorthogonal Principal Skewness Analysis

Abstract

Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a thirdorder generalization of principal component analysis (PCA), its solution of searching for the locally maximum skewness direction is transformed into the problem of calculating the eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, it can prevent the new eigenpairs from converging to the same maxima that has been determined before. However, the eigenvectors of the supersymmetric tensor are not inherently orthogonal in general, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual eigenpairs. In this paper, we propose a new nonorthogonal search strategy to solve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous one, instead of its orthogonal complement space. We give a detailed theoretical proof to illustrate why the new strategy can result in the more accurate eigenpairs. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.

A Parallel Principal Skewness Analysis and Its Application in Radar Target Detection

“…The purpose of BIS is to estimate the mixing matrix, denoted as B . To evaluate the separation performance of the PPSA algorithm, we compare it with algorithms such as FastICA [15], PSA [17], MPSA [18], NPSA [21], and MSDP [25]. In this experiment, this paper selects n grayscale images with a size of 256 × 256 pixels as the source image, where n is 2~6 respectively.…”

“…In order to quantitatively evaluate the performance of the above six algorithms, we use five evaluation indicators to evaluate the separation results obtained by the above six algorithms. The five evaluation indicators are intersymbol interference (ISI) [21], total mean square error (TMSE) [21], correlation coefficient (  ) [21], peak signal-to-noise ratio (PSNR) [21] and running time (T). It is worth noting that except for the PPSA algorithm and the MSDP algorithm, the results obtained by the other four algorithms are random, so we take the average of 10 runs as the final result.…”

“…Thus, except for the first eigenpair, all other solutions obtained by PSA inevitably deviate from the exact eigenpair of the co-skewness tensor. To alleviate this problem, Geng et al [21] proposed a new non-orthogonal PSA (NPSA) algorithm. NPSA offers a unique search strategy by introducing the Kronecker product, which can search for solutions in a larger space and obtain more precise solutions for each eigenpair.…”

The Z-eigenpairs of orthogonally diagonalizable symmetric tensors

“…There are many algorithms and numerous applications that have been investigated concerning this subject. see for example [11,12,13,14,15,16,17,18,19]. In this paper, we mainly investigate the Z-eigenpairs problem of a special class of symmetric tensors, which is orthogonally diagonalizable.…”

Estimating Higher-Order Moments Using Symmetric Tensor Decomposition

“…Tensor decomposition of moment and cumulant tensors are used in a variety of statistical and data science applications, including independent component analysis and blind source separation [10,13,15], clustering [34,12], learning Gaussian mixture models [23,4,20,18,33], latent variable models [2,3], outlier detection [16,1], feature extraction in hyperspectral imagery [19], and multireference alignment [32]. In these cases, it is assumed that the empirical higher-order moment is already computed.…”