Multilinear Low-Rank Tensors on Graphs & Applications

TL;DR

This paper introduces MLRTG, a novel framework combining spectral graph theory and signal processing to analyze low-rank tensors, enabling efficient inverse problem solutions with broad applications.

Contribution

It presents a new graph-based low-rank tensor decomposition and a convex optimization approach for tensor inverse problems, extending classical SVD concepts to graphs.

Findings

MLRTG provides accurate tensor approximations with eigen gap-dependent error bounds.

The framework achieves significant speed-up and low-memory usage.

Applications show improved performance on EEG, FMRI, and other datasets.

Abstract

We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building on this novel decomposition we construct a general class of convex optimization problems for approximately solving low-rank tensor inverse problems, such as tensor Robust PCA. The whole framework is named as 'Multilinear Low-rank tensors on Graphs (MLRTG)'. Our theoretical analysis shows: 1) MLRTG stands on the notion of approximate stationarity of multi-dimensional signals on graphs and 2) the approximation error depends on the eigen gaps of the graphs. We demonstrate applications for a wide variety of 4 artificial and 12 real tensor datasets, such as EEG, FMRI, BCI, surveillance videos and hyperspectral images. Generalization of the tensor concepts to non-euclidean domain, orders of magnitude speed-up, low-memory requirement and significantly enhanced performance at low SNR are the key aspects of our framework.