Chapter 6 - Tensor principal component analysis

Abstract

Tensor principal component analysis (PCA) has attracted increasing attention recently because of its effectiveness in multiway or tensor data analysis. This chapter introduces tensor PCA [1,2] and its variants, including robust tensor PCA (R-TPCA) [2], tensor low-rank representation (TLRR) [3], and outlier robust tensor PCA (OR-TPCA) [4], for handling three kinds of data, namely, (i) Gaussian-noisy tensor data, (ii) sparsely corrupted tensor data, and (iii) outlier-corrupted tensor data. All these methods can exactly recover the clean data of intrinsic low-rank structure in the presence of noise in one setting, with provable performance guarantees. For example, for low-rank tensor data with sparse corruptions, R-TPCA and TLRR can exactly recover the clean data under mild conditions, while TLRR can exactly retrieve their true tensor subspaces and hence cluster them accurately. Besides, the objective function of these methods can be optimized via efficient convex programming with convergence guarantees. Finally, we also evaluate these methods on real applications, such as image/video recovery, outlier detection, and face clustering and classification.