Exact clustering in tensor block model: Statistical optimality and computational limit

High‐order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non‐convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, high‐order Lloyd algorithm (HLloyd), and high‐order spectral clustering (HSC), for high‐order clustering. The convergence guarantees and statistical optimality are established for the proposed procedure under a mild sub‐Gaussian noise assumption. Under the Gaussian tensor block model, we completely characterise the statistical‐computational trade‐off for achieving high‐order exact clustering based on three different signal‐to‐noise ratio regimes. The analysis relies on new techniques of high‐order spectral perturbation analysis and a ‘singular‐value‐gap‐free’ error bound in tensor estimation, which are substantially different from the matrix spectral analyses in the literature. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets.