Meta-factorization

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 Nystroem 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.

Reference Paper

Karpowicz, M. P. (2021). A theory of meta-factorization. arXiv preprint arXiv:2111.14385.

@misc{karpowicz2021theory,
      title={A theory of meta-factorization}, 
      author={Michał P. Karpowicz},
      year={2021},
      eprint={2111.14385},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}