A Nested Matrix-Tensor Model for Noisy Multi-view Clustering
This work addresses multi-view clustering problems where data has non-uniform noise across views, offering a tensor-based method that improves accuracy over simpler approaches, though it is incremental as it builds on existing spiked tensor models.
The paper tackles multi-view clustering with noisy observations by proposing a nested matrix-tensor model that extends the spiked rank-one tensor model, showing that a best rank-one tensor approximation can estimate hidden clusters with theoretical guarantees and better accuracy compared to naive unfolding-based methods.
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.