Generalized Inverses of Matrix Products: From Fundamental Subspaces to Randomized Decompositions
This work addresses foundational linear algebra problems for researchers in randomized algorithms and matrix computations, offering incremental theoretical insights into existing methods.
The paper tackles the problem of generalizing and randomizing matrix inverses for products, establishing a unifying framework based on fundamental subspaces and deriving new formulas, including a generalized randomized inverse that equals the true inverse under rank-preserving sketching, with applications in sparse sensor placement and effective resistance estimation where it provides rigorous error bounds.
We investigate the Moore-Penrose pseudoinverse and generalized inverse of a matrix product $A=CR$ to establish a unifying framework for generalized and randomized matrix inverses. This analysis is rooted in first principles, focusing on the geometry of the four fundamental subspaces. We examine: (1) the reverse order law, $A^+ = R^+C^+$, which holds when $C$ has independent columns and $R$ has independent rows, (2) the universally correct formula, $A^+ = (C^+CR)^+(CRR^+)^+$, providing a geometric interpretation of the mappings between the involved subspaces, (3) a new generalized randomized formula, $A^+_p = (P^TA)^+P^TAQ(AQ)^+$, which gives $A^+_p = A^+$ if and only if the sketching matrices $P$ and $Q$ preserve the rank of $A$, i.e., $\mathrm{rank}(P^TA) = \mathrm{rank}(AQ) = \mathrm{rank}(A)$. The framework is extended to generalized $\{1,2\}$-inverses and specialized forms, revealing the underlying structure of established randomized linear algebra algorithms, including randomized SVD, the Nyström approximation, and CUR decomposition. We demonstrate applications in sparse sensor placement and effective resistance estimation. For the latter, we provide a rigorous quantitative analysis of an approximation scheme, establishing that it always underestimates the true resistance and deriving a worst-case spectral bound on the error of resistance differences.