Seriation of Toeplitz and latent position matrices: optimal rates and computational trade-offs

Clément Berenfeld; Alexandra Carpentier; Nicolas Verzelen

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arXiv · 2408.10004

Seriation of Toeplitz and latent position matrices: optimal rates and computational trade-offs

Clément Berenfeld·Alexandra Carpentier·Nicolas Verzelen

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Abstract

In this paper, we consider the problem of seriation of a permuted structured matrix based on noisy observations. The entries of the matrix relate to an expected quantification of interaction between two objects: the higher the value, the closer the objects. A popular structured class for modelling such matrices is the permuted Robinson class, namely the set of matrices whose coefficients are decreasing away from its diagonal, up to a permutation of its lines and columns. We consider in this paper two submodels of Robinson matrices: the Toeplitz model, and the latent position model. We provide a computational lower bound based on the low-degree paradigm, which hints that there is a statistical-computational gap for seriation when measuring the error based on the Frobenius norm. We also provide a simple and polynomial-time algorithm that achives this lower bound. Along the way, we also characterize the information-theory optimal risk thereby giving evidence for the extent of the computation/information gap for this problem.

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Clément Berenfeld, Alexandra Carpentier, Nicolas Verzelen. 2024-08-19. Seriation of Toeplitz and latent position matrices: optimal rates and computational trade-offs. https://arxiv.org/abs/2408.10004

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