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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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Provable Convex Co-clustering of Tensors
Cluster analysis is a fundamental tool for pattern discovery of complex heterogeneous data. Prevalent clustering methods mainly focus on vector or matrix-variate data and are not applicable to general-order tensors, which arise frequently in modern scientific and business applications. Moreover, there is a gap between statistical guarantees and computational efficiency for existing tensor clustering solutions due to the nature of their non-convex formulations. In this work, we bridge this gap by developing a provable convex formulation of tensor co-clustering. Our convex co-clustering (CoCo) estimator enjoys stability guarantees and its computational and storage costs are polynomial in the size of the data. We further establish a non-asymptotic error bound for the CoCo estimator, which reveals a surprising ``blessing of dimensionality" phenomenon that does not exist in vector or matrix-variate cluster analysis. Our theoretical findings are supported by extensive simulated studies. Finally, we apply the CoCo estimator to the cluster analysis of advertisement click tensor data from a major online company. Our clustering results provide meaningful business insights to improve advertising effectiveness.
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- PAR ID:
- 10199582
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 21
- Issue:
- 214
- ISSN:
- 1533-7928
- Page Range / eLocation ID:
- 1-58
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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