We introduce 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 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:我们引入了元因子化,这个理论将矩阵分解描述为线性矩阵方程式的解决方案:投影器和重建方程式。元因子化重建了已知的因子化,揭示了内部结构,并允许采用SVD、QR和UTV的因子化所显示的修改。元因子化的前景也为通用矩阵反向和随机线性线性代数算法的计算方面提供了洞察力。摩尔-彭罗斯伪反射法、普遍Nystr\"{o}m法和CUR分解法之间的关系在这里作为例证。最后,元因子化提供了新因子化结构的提示,并提供了创造新因子化的潜力。