Robust block tensor PCA with F-norm projection framework

Abstract

Tensor principal component analysis (TPCA), also known as Tucker decomposition, ensures that the extracted “core tensor” maximizes the variance of the sample projections. Nevertheless, this method is particularly susceptible to noise and outliers. This is due to the utilization of the squared F-norm as the distance metric. In addition, it lacks constraints on the discrepancies between the original tensors and the projected tensors. To address these issues, a novel tensor-based trigonometric projection framework is proposed using F-norm to measure projection distances. Tensor data are first processed utilizing a blocking recombination technique prior to projection, thus enhancing the representation of the data at a local spatio-temporal level. Then, we present a block TPCA with the F-norm metric (BTPCA-F) and develop an iterative greedy algorithm for solving BTPCA-F. Subsequently, regarding the F-norm projection relation as the “Pythagorean Theorem”, we provide three different objective functions, namely, the tangent, cosine and sine models. These three functions directly or indirectly achieve the two objectives of maximizing projection distances and minimizing reconstruction errors. The corresponding tangent, cosine and sine solution algorithms based on BTPCA-F (called tan-BTPCA-F, cos-BTPCA-F and sin-BTPCA-F) are presented to optimize the objective functions, respectively. The convergence and rotation invariance of these algorithms are rigorously proved theoretically and discussed in detail. Lastly, extensive experimental results illustrate that the proposed methods significantly outperform the existing TPCA and the related 2DPCA algorithms.