A theory of meta-factorization

Abstract

Meta-factorization asks how different factors and transforms describe the same matrix and how a factor can be replaced while preserving that matrix. Projector and reconstruction equations give a common description. For $A=FG{H}^{\ast }$ with outer factors of full column rank $k$, $Y^{\ast }F=H^{\ast }X=I_k$ certifies generally oblique projectors. Exact reconstruction requires their ranges to contain the column spaces of $A$ and $A^{\ast }$. Two design matrices parametrize every certifying pair; the core is unique and nonsingular at $k=\mathrm{rank}(A)$. Detailed constructions recover SVD, column-pivoted QR, UTV, generalized Nystrom, CUR and classical pseudoinverse formulas with their structural hypotheses. The framework makes factor replacement testable from stored factors. For a supplied candidate $\check{F}$ and transform ${\check{Y}}^{\ast }$, with $H$ of full column rank, $\check{G}=({\check{Y}}^{\ast }F)G$ preserves $A$ exactly when $[F-\check{F}({\check{Y}}^{\ast }F)]G=0$. Test, repair and exact factored residual require no access to $A$. With the pseudoinverse as transform, the updated product is a closest reconstruction within the candidate's column space in every unitarily invariant norm. We characterize the accepted subspaces under the stated inverse and rank conditions and extend replacement to different stored widths. Nonsingular generalized right-hand sides change certifying coordinates; finite-order choices yield potent operators and periodic reconstruction. Classical Fourier components test preservation of the core at every step, including singular cores. The contribution is a common description with explicit replacement and reconstruction consequences. Numerical acceptance, conditioning and candidate selection require further analysis.

New sections added, new illustrations, major revision and debugging, new matlab code for demonstrations

References in corpus (2)

• Improving CUR Matrix Decomposition and the Nyström Approximation via Adaptive Sampling
• Lecture Notes on Randomized Linear Algebra