  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.
