Seriation of Tœplitz and latent position matrices: Optimal rates and computational trade-offs

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 modeling 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 rows and columns. We consider in this paper two submodels of Robinson matrices: the Tœplitz 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 achieves 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.