Jointly Modeling and Clustering Tensors in High Dimensions

• Activity Number: 234 - New Challenges in Statistical Learning and Inference for Complex Data
• Type: Topic Contributed
• Date/Time: Tuesday, August 9, 2022 : 8:30 AM to 10:20 AM
• Sponsor: Section for Statistical Programmers and Analysts
• Author(s): Biao Cai* and Emma Jingfei Zhang and Will Wei Sun
• Companies: Yale University and University of Miami and Purdue University
• Keywords: expectation conditional maximization; computational and statistical errors; tensor clustering; tensor decomposition; unsupervised learning

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 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 is demonstrated by simulation and one real data application.