A Nested Matrix-Tensor Model for Noisy Multi-view Clustering
Mohamed El Amine Seddik Mastane Achab Henrique Goulart Merouane Debbah
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.