Graph Regularized Low-Rank Tensor-Train for Robust Principal Component Analysis

Abstract

With the advance of sensor technology, it is becoming more commonplace to collect multi-mode data, i.e., tensors, with high dimensionality. To deal with the large amounts of redundancy in tensorial data, different dimensionality reduction methods such as low-rank tensor decomposition have been developed. While low-rank decompositions capture the global structure, there is a need to leverage the underlying local geometry through manifold learning methods. Manifold learning methods have been widely considered in tensor factorization to incorporate the low-dimensional geometry of the underlying data. However, existing techniques focus on only one mode of the data and exploit correlations among the features to reduce the dimension of the feature vectors. Recently, multiway graph signal processing approaches that exploit the correlations among all modes of a tensor have been proposed to learn low-dimensional representations. Inspired by this idea, in this letter we propose a graph regularized robust tensor-train decomposition method where the graph regularization is applied across each mode of the tensor to incorporate the local geometry. As the resulting optimization problem is computationally prohibitive due to the high dimensionality of the graph regularization terms, an equivalence between mode- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> canonical unfolding and regular mode- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> unfolding is derived resulting in a computationally efficient optimization algorithm. The proposed method is evaluated on both synthetic and real tensors for denoising and tensor completion.