We consider multilevel low rank (MLR) matrices, defined as a row and column permutation of a sum of matrices, each one a block diagonal refinement of the previous one, with all blocks low rank given in factored form. MLR matrices extend low rank matrices but share many of their properties, such as the total storage required and complexity of matrix-vector multiplication. We address three problems that arise in fitting a given matrix by an MLR matrix in the Frobenius norm. The first problem is factor fitting, where we adjust the factors of the MLR matrix. The second is rank allocation, where we choose the ranks of the blocks in each level, subject to the total rank having a given value, which preserves the total storage needed for the MLR matrix. The final problem is to choose the hierarchical partition of rows and columns, along with the ranks and factors. This paper is accompanied by an open source package that implements the proposed methods.

翻译：我们考虑了多级低秩（MLR）矩阵，这类矩阵定义为行与列的排列后多个矩阵之和，其中每个矩阵都是前一个矩阵的分块对角精化，所有块均以分解形式给出且具有低秩性。MLR矩阵扩展了低秩矩阵的概念，同时保留了其许多特性，例如所需的总存储量以及矩阵-向量乘法的复杂度。本文解决了在Frobenius范数下用MLR矩阵拟合给定矩阵时出现的三个问题。第一个问题是因子拟合，即调整MLR矩阵的因子。第二个问题是秩分配，即在给定总秩数值的条件下选择每一级中各块的秩，该条件保持了MLR矩阵所需的总存储量。最后一个问题则需选择行与列的层次划分，同时确定相应的秩与因子。本文附有实现所提方法的开源软件包。