Nonparametric tensor estimation with unknown permutations

We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation system, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensors up to arbitrarily index permutations; the model incorporates the popular block models and graphon models. We show that a constrained least-squares estimate in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree of $m(m-1)/2$ is sufficient for accurate recovery of order-$m$ tensors, whereas higher smoothness exhibits no further benefits. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime date analysis.