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In this paper, we propose a nested matrix-tensor model which extends thespiked rank-one tensor model of order three. This model is particularlymotivated by a multi-view clustering problem in which multiple noisyobservations of each data point are acquired, with potentially non-uniformvariances along the views. In this case, data can be naturally represented byan order-three tensor where the views are stacked. Given such a tensor, weconsider the estimation of the hidden clusters via performing a best rank-onetensor approximation. In order to study the theoretical performance of thisapproach, we characterize the behavior of this best rank-one approximation interms of the alignments of the obtained component vectors with the hidden modelparameter vectors, in the large-dimensional regime. In particular, we show thatour theoretical results allow us to anticipate the exact accuracy of theproposed clustering approach. Furthermore, numerical experiments indicate thatleveraging our tensor-based approach yields better accuracy compared to a naiveunfolding-based algorithm which ignores the underlying low-rank tensorstructure. Our analysis unveils unexpected and non-trivial phase transitionphenomena depending on the model parameters, ``interpolating'' between thetypical behavior observed for the spiked matrix and tensor models.

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