Jointly Modeling and Clustering Tensors in High Dimensions

TL;DR

This paper introduces a high-dimensional tensor mixture model with an efficient HECM algorithm for clustering tensors, leveraging tensor structures like low-rankness and separability, with proven convergence and improved accuracy.

Contribution

The paper develops a novel high-dimensional tensor mixture model with an efficient EM-based algorithm that handles non-convexity and demonstrates theoretical convergence and practical effectiveness.

Findings

The HECM algorithm converges geometrically under proper initialization.

The method achieves higher clustering accuracy than existing benchmarks.

Numerical experiments and a medical study validate the approach.

Abstract

We consider the problem of jointly modeling and clustering populations of tensors by introducing a high-dimensional tensor mixture model with heterogeneous covariances. To effectively tackle the high dimensionality of tensor objects, we employ plausible dimension reduction assumptions that exploit the intrinsic structures of tensors such as low-rankness in the mean and separability in the covariance. In estimation, we develop an efficient high-dimensional expectation-conditional-maximization (HECM) algorithm that breaks the intractable optimization in the M-step into a sequence of much simpler conditional optimization problems, each of which is convex, admits regularization and has closed-form updating formulas. Our theoretical analysis is challenged by both the non-convexity in the EM-type estimation and having access to only the solutions of conditional maximizations in the M-step, leading to the notion of dual non-convexity. We demonstrate that the proposed HECM algorithm, with an appropriate initialization, converges geometrically to a neighborhood that is within statistical precision of the true parameter. The efficacy of our proposed method is demonstrated through comparative numerical experiments and an application to a medical study, where our proposal achieves an improved clustering accuracy over existing benchmarking methods.