Robust tensor train component analysis

Abstract

Summary Robust Principal Component Analysis plays a key role in various fields such as image and video processing, data mining, and hyperspectral data analysis. In this paper, we study the problem of robust tensor train (TT) principal component analysis from partial observations, which aims to decompose a given tensor into the low TT rank and sparse components. The decomposition of the proposed model is used to find the hidden factors and help alleviate the curse of dimensionality via a set of connected low‐rank tensors. A relaxation model is to minimize a weighted combination of the sum of nuclear norms of unfolding matrices of core tensors and the tensor ℓ1 norm. A proximal alternating direction method of multipliers is developed to solve the resulting model. Furthermore, we show that any cluster point of the convergent subsequence is a Karush‐Kuhn‐Tucker point of the proposed model under some conditions. Extensive numerical examples on both synthetic data and real‐world datasets are presented to demonstrate the effectiveness of the proposed approach.

An efficient tensor regression for high-dimensional data

“…Modern society has witnessed an enormous progress in collecting all kinds of data with complex structures, and datasets in the form of tensors have been increasingly encountered from many fields, such as signal processing (Zhao et al, 2012;Shimoda et al, 2012), medical imaging analysis (Zhou et al, 2013;Li et al, 2018), economics and finance (Chen et al, 2022;Wang et al, 2021b), digital marketing (Hao et al, 2021;Bi et al, 2018) and many others. Many unsupervised learning methods have been considered for these datasets, and they include the principal component analysis (Zhang and Ng, 2022), clustering (Sun and Li, 2019;Luo and Zhang, 2022), and factor modeling (Bi et al, 2018;Chen et al, 2022). On the other hand, there is a bigger literature for analyzing tensor-valued observations with supervised learning methods, and most of them come from the area of machine learning by using neural networks (Novikov et al, 2015;Kossaifi et al, 2020).…”