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.
翻译:本文考虑了矩阵变差后勤回归的问题,通过获得最低负载风险的下限,得出了估算物流回归问题中低位系数矩阵的基本差错阈值,明确取决于共差的大小和分布、系数矩阵的等级和能量以及样本的数量,由此产生的约束与问题的内在自由程度成正比,这表明低位矩阵物流回归问题的抽样复杂性可能低于病媒化物流回归问题。 这项工作中使用的验证技术也为开发单位变差后勤回归问题的最低下限奠定了基础。