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We propose a new framework for the analysis of low-rank tensors which lies atthe intersection of spectral graph theory and signal processing. As a firststep, we present a new graph based low-rank decomposition which approximatesthe classical low-rank SVD for matrices and multi-linear SVD for tensors. Then,building on this novel decomposition we construct a general class of convexoptimization problems for approximately solving low-rank tensor inverseproblems, such as tensor Robust PCA. The whole framework is named as'Multilinear Low-rank tensors on Graphs (MLRTG)'. Our theoretical analysisshows: 1) MLRTG stands on the notion of approximate stationarity ofmulti-dimensional signals on graphs and 2) the approximation error depends onthe eigen gaps of the graphs. We demonstrate applications for a wide variety of4 artificial and 12 real tensor datasets, such as EEG, FMRI, BCI, surveillancevideos and hyperspectral images. Generalization of the tensor concepts tonon-euclidean domain, orders of magnitude speed-up, low-memory requirement andsignificantly enhanced performance at low SNR are the key aspects of ourframework.

Multilinear Low-Rank Tensors on Graphs & Applications