The matrix normal model, the family of Gaussian matrix-variate distributions whose covariance matrix is the Kronecker product of two lower dimensional factors, is frequently used to model matrix-variate data. The tensor normal model generalizes this family to Kronecker products of three or more factors. We study the estimation of the Kronecker factors of the covariance matrix in the matrix and tensor models. We show nonasymptotic bounds for the error achieved by the maximum likelihood estimator (MLE) in several natural metrics. In contrast to existing bounds, our results do not rely on the factors being well-conditioned or sparse. For the matrix normal model, all our bounds are minimax optimal up to logarithmic factors, and for the tensor normal model our bound for the largest factor and overall covariance matrix are minimax optimal up to constant factors provided there are enough samples for any estimator to obtain constant Frobenius error. In the same regimes as our sample complexity bounds, we show that an iterative procedure to compute the MLE known as the flip-flop algorithm converges linearly with high probability. Our main tool is geodesic strong convexity in the geometry on positive-definite matrices induced by the Fisher information metric. This strong convexity is determined by the expansion of certain random quantum channels. We also provide numerical evidence that combining the flip-flop algorithm with a simple shrinkage estimator can improve performance in the undersampled regime.
翻译:矩阵正态模型, Gaussian 矩阵变异分布的组合, 其共差矩阵是两个较低维度因素的Kronecker 产物, 通常用于模拟矩阵变异数据。 强度正常模型将这个家族向Kronecker 产物概括为三个或三个以上因素的Kronecker 产物。 我们研究矩阵和振幅模型中共差矩阵矩阵矩阵变异分布的Kronecker因素的估计。 我们在若干自然度指标中展示了最大可能性估测仪(MLE)所实现的错误的非随机边框。 与现有的界限不同, 我们的结果并不依赖于条件良好或稀释的因素。 对于矩阵正常模型模型, 我们所有的细度模型将这个家族对Kroneckererer 的全称对三个或三个以上因素进行概括化。 我们研究了矩阵变异性矩阵的Kronecker值系数和常数系数。 我们的精度缩略度缩略度系统可以提供相同的系统, 与我们所选的精度缩缩缩缩缩缩缩缩缩缩图相比, 我们的比程序将MLE 的精度缩缩缩缩缩缩缩缩缩缩缩缩缩缩缩缩缩缩缩为正数, 由我们所认识为正缩缩缩缩缩缩的精确的精确度为正缩略图的精确度, 的缩略度为正缩略度为正缩缩略度为正缩缩缩缩缩缩缩缩缩缩缩略图。