Performance Gaps in Multi-view Clustering under the Nested Matrix-Tensor Model

Hugo Lebeau, Mohamed El Amine Seddik, José Henrique De Morais Goulart

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Keywords: random tensors, multi-view clustering, random matrix theory, tensor unfolding

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TL;DR: In a multi-view clustering context, the performance gap between a tensor unfolding approach and and a purely tensor-based method are quantified, relying on a nested matrix-tensor model and tools from random matrix theory.

Abstract: We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically examined the performance of a tensor-based approach, which relies on finding a best rank-one approximation, a problem known to be computationally hard. A tractable alternative approach consists in computing instead the best rank-one (matrix) approximation of an unfolding of the observed tensor data, but its performance was hitherto unknown. We quantify here the performance gap between these two approaches, in particular by deriving the precise algorithmic threshold of the unfolding approach and demonstrating that it exhibits a BBP-type transition behavior. This work is therefore in line with recent contributions which deepen our understanding of why tensor-based methods surpass matrix-based methods in handling structured tensor data.

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Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Submission Number: 1342