# Jointly Modeling and Clustering Tensors in High Dimensions

arXiv: https://arxiv.org/abs/2104.07773 · HTML: https://mlsanity.com/paper/2104.07773/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 56 references that are in the map.
- First posted on arXiv 15 Apr 2021.

## Builds on (most cited first, 25 of 25)

- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Singularity, Misspecification, and the Convergence Rate of EM](https://mlsanity.com/paper/1810.00828/) (2018)
- [Nonconvex Low-Rank Tensor Completion from Noisy Data](https://mlsanity.com/paper/1911.04436/) (2019)
- [Tensor Clustering with Planted Structures: Statistical Optimality and Computational Limits](https://mlsanity.com/paper/2005.10743/) (2020)
- [On the Minimax Optimality of the EM Algorithm for Learning Two-Component Mixed Linear Regression](https://mlsanity.com/paper/2006.02601/) (2020)
- [Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entries](https://mlsanity.com/paper/1711.04934/) (2017)
- [Parsimonious Tensor Response Regression](https://mlsanity.com/paper/1501.07815/) (2015)
- [Regularized EM Algorithms: A Unified Framework and Statistical Guarantees](https://mlsanity.com/paper/1511.08551/) (2015)
- [Provable Sparse Tensor Decomposition](https://mlsanity.com/paper/1502.01425/) (2015)
- [Tensor Graphical Lasso (TeraLasso)](https://mlsanity.com/paper/1705.03983/) (2017)
- [Multilayer tensor factorization with applications to recommender systems](https://mlsanity.com/paper/1711.01598/) (2017)
- [Provable Convex Co-clustering of Tensors](https://mlsanity.com/paper/1803.06518/) (2018)
- [Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality](https://mlsanity.com/paper/2006.08580/) (2020)
- [Dynamic Tensor Clustering](https://mlsanity.com/paper/1708.07259/) (2017)
- [Sparse and Low-rank Tensor Estimation via Cubic Sketchings](https://mlsanity.com/paper/1801.09326/) (2018)
- [Cross: Efficient Low-rank Tensor Completion](https://mlsanity.com/paper/1611.01129/) (2016)
- [Simultaneous Clustering and Estimation of Heterogeneous Graphical Models](https://mlsanity.com/paper/1611.09391/) (2016)
- [Covariate-Adjusted Tensor Classification in High-Dimensions](https://mlsanity.com/paper/1805.04421/) (2018)
- [A Doubly-Enhanced EM Algorithm for Model-Based Tensor Clustering](https://mlsanity.com/paper/2012.10032/) (2020)
- [Effective Tensor Sketching via Sparsification](https://mlsanity.com/paper/1710.11298/) (2017)
- [Sparse Tensor Additive Regression](https://mlsanity.com/paper/1904.00479/) (2019)
- [Partially Observed Dynamic Tensor Response Regression](https://mlsanity.com/paper/2002.09735/) (2020)
- [Generalized Connectivity Matrix Response Regression with Applications in Brain Connectivity Studies](https://mlsanity.com/paper/1810.03192/) (2018)
- [Clustering Higher Order Data: An Application to Pediatric Multi-variable Longitudinal Data](https://mlsanity.com/paper/1907.08566/) (2019)

## Built on by (most cited first, 1 of 1)

- [Data Understanding Survey: Pursuing Improved Dataset Characterization Via Tensor-based Methods](https://mlsanity.com/paper/2510.14161/) (2025)

## Abstract (arXiv)

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.