This paper introduces meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with the example of SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.
翻译:本文介绍元因素,这是一个将矩阵分解描述为线性矩阵方程式解决方案的理论:投影器和重建方程式。元因素重组了已知的因子化,揭示了内部结构,并允许进行修改,如SVD、QR和UTV因子化的例子所示。元因素化的前景也提供了对通用矩阵反向和随机线性线性代数算法的计算方面的洞察。Moore-Penrose伪反射法、普遍的Nystr\\'{o}m法和CUR分解法之间的关系在这里作为例证。最后,元因素化提供了新因子化结构的提示,并提供了创建新因子化法的潜力。