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

TL;DR

This paper establishes a fundamental lower bound on the estimation error for low-rank matrix coefficients in matrix-variate logistic regression, highlighting the intrinsic difficulty and sample complexity of the problem.

Contribution

It derives the first minimax lower bound for low-rank matrix logistic regression, explicitly relating error thresholds to problem dimensions, rank, and data distribution.

Findings

Lower bound proportional to intrinsic degrees of freedom

Sample complexity can be lower than vectorized logistic regression

Proof techniques applicable to tensor-variate logistic regression

Abstract

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.