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：我们研究多级低秩（MLR）矩阵，其定义为一系列矩阵之和的行列置换，其中每个矩阵都是前一个矩阵的块对角细化，且所有块均以分解形式给出的低秩矩阵。MLR矩阵扩展了低秩矩阵的概念，但保留了其诸多特性，例如所需的总存储量和矩阵-向量乘法的计算复杂度。我们针对在Frobenius范数下用MLR矩阵拟合给定矩阵时产生的三个问题展开研究。第一个问题是因子拟合，即调整MLR矩阵的分解因子。第二个问题是秩分配，即在保持MLR矩阵所需总存储量不变的前提下，为每层中的块选择秩（受限于总秩为给定值）。最后的问题是如何选择行列的层次化分区，以及相应的秩和因子。本文附有一个实现所提方法的开源软件包。