Factor Fitting, Rank Allocation, and Partitioning in Multilevel Low Rank Matrices
Tetiana Parshakova, Trevor Hastie, Eric Darve, Stephen Boyd
我们考虑多级低等级(MLR)矩阵,定义为矩阵总和的行和列排列,每个矩阵的块对角细化,所有块低等级以因子形式给出。 MLR矩阵扩展了低等级矩阵,但共享其许多属性,例如所需的总存储和矩阵向量乘法的复杂性。 我们解决了在 Frobenius 规范中通过 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,我们调整MLR矩阵的因子。 第二个是等级分配,我们选择每个级别中的块的等级,但总排名具有给定值,保留MLR矩阵所需的总存储。 最后的问题是选择行和列的分层分区,以及等级和因素。 本文附有一个开源包,实现了建议的方法。
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 fa...