Multilinear Low-Rank Tensors on Graphs & Applications
This work addresses tensor inverse problems for applications like EEG and surveillance videos, offering a novel method with broad improvements in efficiency and robustness.
The authors tackled the problem of low-rank tensor analysis by proposing a new framework called Multilinear Low-Rank Tensors on Graphs (MLRTG), which integrates spectral graph theory and signal processing to achieve orders of magnitude speed-up, low-memory requirements, and significantly enhanced performance at low SNR across 16 datasets.
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.