# Robust Regularized Low-Rank Matrix Models for Regression and Classification

arXiv: https://arxiv.org/abs/2205.07106 · HTML: https://mlsanity.com/paper/2205.07106/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 53 references that are in the map.
- First posted on arXiv 14 May 2022.

## Builds on (most cited first, 9 of 9)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Tucker Tensor Regression and Neuroimaging Analysis](https://mlsanity.com/paper/1304.5637/) (2013)
- [A Shrinkage Principle for Heavy-Tailed Data: High-Dimensional Robust Low-Rank Matrix Recovery](https://mlsanity.com/paper/1603.08315/) (2016)
- [Selecting the number of principal components: estimation of the true rank of a noisy matrix](https://mlsanity.com/paper/1410.8260/) (2014)
- [Robust PCA by Manifold Optimization](https://mlsanity.com/paper/1708.00257/) (2017)
- [Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-order Convergence](https://mlsanity.com/paper/2011.08360/) (2020)
- [Robust Reduced Rank Regression](https://mlsanity.com/paper/1509.03938/) (2015)
- [Analysis of Generalized Bregman Surrogate Algorithms for Nonsmooth Nonconvex Statistical Learning](https://mlsanity.com/paper/2112.09191/) (2021)
- [Near-Optimal Estimation of Simultaneously Sparse and Low-Rank Matrices from Nested Linear Measurements](https://mlsanity.com/paper/1506.08159/) (2015)

## Built on by (most cited first, 2 of 2)

- [Optimal Clustering by Lloyd Algorithm for Low-Rank Mixture Model](https://mlsanity.com/paper/2207.04600/) (2022)
- [Online Tensor Learning: Computational and Statistical Trade-offs, Adaptivity and Optimal Regret](https://mlsanity.com/paper/2306.03372/) (2023)

## Abstract (arXiv)

While matrix variate regression models have been studied in many existing works, classical statistical and computational methods for the analysis of the regression coefficient estimation are highly affected by high dimensional and noisy matrix-valued predictors. To address these issues, this paper proposes a framework of matrix variate regression models based on a rank constraint, vector regularization (e.g., sparsity), and a general loss function with three special cases considered: ordinary matrix regression, robust matrix regression, and matrix logistic regression. We also propose an alternating projected gradient descent algorithm. Based on analyzing our objective functions on manifolds with bounded curvature, we show that the algorithm is guaranteed to converge, all accumulation points of the iterates have estimation errors in the order of O(1/√n) asymptotically and substantially attaining the minimax rate. Our theoretical analysis can be applied to general optimization problems on manifolds with bounded curvature and can be considered an important technical contribution to this work. We validate the proposed method through simulation studies and real image data examples.