🤖 AI Summary
This work proposes a unified framework for understanding generalized and randomized inverses of matrix products, revealing their intrinsic structure in subspace geometry and randomized linear algebra. Building upon the geometric relationships among the four fundamental subspaces, we develop a cohesive formulation encompassing the Moore–Penrose pseudoinverse, {1,2}-inverses, and their randomized counterparts, yielding both a universally valid generalized inverse expression and novel randomized inverse representations. The framework elucidates the common structural foundation underlying algorithms such as randomized SVD, Nyström approximation, and CUR decomposition. Moreover, it establishes, for the first time, rigorous error bounds for effective resistance estimation, proving that the error is always an underestimate and providing worst-case spectral bounds. These theoretical insights are applicable to practical problems including sparse sensor placement.
📝 Abstract
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 Nystr\"om 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.