We consider the problem of multiway clustering in the presence of unknown degree heterogeneity. Such data problems arise commonly in applications such as recommendation systems, neuroimaging, community detection, and hypergraph partitions in social networks. The allowance of degree heterogeneity provides great flexibility in clustering models, but the extra complexity poses significant challenges in both statistics and computation. Here, we develop a degree-corrected tensor block model with estimation accuracy guarantees. We present the phase transition of clustering performance based on the notion of angle separability, and we characterize three signal-to-noise regimes corresponding to different statistical-computational behaviors. In particular, we demonstrate that an intrinsic statistical-to-computational gap emerges only for tensors of order three or greater. Further, we develop an efficient polynomial time algorithm that provably achieves exact clustering under mild signal conditions. The efficacy of our procedure is demonstrated through both simulations and analyses of Peru Legislation dataset.
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Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit
Abstract 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.
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- Award ID(s):
- 2203741
- NSF-PAR ID:
- 10410239
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 84
- Issue:
- 5
- ISSN:
- 1369-7412
- Page Range / eLocation ID:
- 1666 to 1698
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1111/rssb.12547
