# A Minimax Lower Bound for Low-Rank Matrix-Variate Logistic Regression

arXiv: https://arxiv.org/abs/2105.14673 · HTML: https://mlsanity.com/paper/2105.14673/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 23 references that are in the map.
- First posted on arXiv 31 May 2021.

## Builds on (most cited first, 7 of 7)

- [Logistic Regression: The Importance of Being Improper](https://mlsanity.com/paper/1803.09349/) (2018)
- [High-dimensional classification by sparse logistic regression](https://mlsanity.com/paper/1706.08344/) (2017)
- [ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching](https://mlsanity.com/paper/1911.03804/) (2019)
- [Sparse Bilinear Logistic Regression](https://mlsanity.com/paper/1404.4104/) (2014)
- [On the Minimax Risk of Dictionary Learning](https://mlsanity.com/paper/1507.05498/) (2015)
- [Learning Product Graphs Underlying Smooth Graph Signals](https://mlsanity.com/paper/2002.11277/) (2020)
- [Minimax Lower Bounds for Kronecker-Structured Dictionary Learning](https://mlsanity.com/paper/1605.05284/) (2016)

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

- [Structured Low-Rank Tensors for Generalized Linear Models](https://mlsanity.com/paper/2308.02922/) (2023)

## Abstract (arXiv)

This paper considers the problem of matrix-variate logistic regression. It derives the fundamental error threshold on estimating low-rank coefficient matrices in the logistic regression problem by obtaining a lower bound on the minimax risk. The bound depends explicitly on the dimension and distribution of the covariates, the rank and energy of the coefficient matrix, and the number of samples. The resulting bound is proportional to the intrinsic degrees of freedom in the problem, which suggests the sample complexity of the low-rank matrix logistic regression problem can be lower than that for vectorized logistic regression. The proof techniques utilized in this work also set the stage for development of minimax lower bounds for tensor-variate logistic regression problems.