A Nested Matrix-Tensor Model for Noisy Multi-view Clustering

• URL: http://arxiv.org/abs/2305.19992v1
• Date: Wed, 31 May 2023 16:13:46 GMT
• Title: A Nested Matrix-Tensor Model for Noisy Multi-view Clustering
• Authors: Mohamed El Amine Seddik, Mastane Achab, Henrique Goulart, Merouane Debbah
• Abstract summary: We propose a nested matrix-tensor model which extends the spiked rank-one tensor model of order three. We show that our theoretical results allow us to anticipate the exact accuracy of the proposed clustering approach. Our analysis unveils unexpected and non-trivial phase transition phenomena depending on the model parameters.
• Score: 5.132856740094742
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we propose a nested matrix-tensor model which extends the spiked rank-one tensor model of order three. This model is particularly motivated by a multi-view clustering problem in which multiple noisy observations of each data point are acquired, with potentially non-uniform variances along the views. In this case, data can be naturally represented by an order-three tensor where the views are stacked. Given such a tensor, we consider the estimation of the hidden clusters via performing a best rank-one tensor approximation. In order to study the theoretical performance of this approach, we characterize the behavior of this best rank-one approximation in terms of the alignments of the obtained component vectors with the hidden model parameter vectors, in the large-dimensional regime. In particular, we show that our theoretical results allow us to anticipate the exact accuracy of the proposed clustering approach. Furthermore, numerical experiments indicate that leveraging our tensor-based approach yields better accuracy compared to a naive unfolding-based algorithm which ignores the underlying low-rank tensor structure. Our analysis unveils unexpected and non-trivial phase transition phenomena depending on the model parameters, ``interpolating'' between the typical behavior observed for the spiked matrix and tensor models.

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