Researchers have proved that for any real matrix of rank r, a simple local‑search procedure can generate an ah‑symmetric reflexive generalized inverse that uses no more than k rows and whose 2‑1‑norm is within a factor that depends only on the ratio k/r to the optimum. The algorithm, equivalent to the classic Fedorov exchange algorithm for D‑optimal design, repeatedly swaps rows to maximize the squared rank‑r volume of the chosen columns. When the swap tolerance ε is zero, the loss factor equals r when k=r, and drops below √(2r) whenever k≥2r. For ε=1/k the method can be run in polynomial time on rational data.

The main result gives a concrete bound: the loss of restricting the inverse to a k‑subset K is at most √{k(r+εk)/(k−r+1)}. This certificate is tight up to a constant that depends only on k/r, and for many instances it matches the best possible bound of √r for supports of a given size.

Experiments on benchmark matrices (S2 and S4) show that the practical 2‑1‑norm performance of the volume‑local search is far better than the worst‑case guarantee. In these tests the method produced 2‑1‑norms up to 27 % smaller than two state‑of‑the‑art ADMM heuristics, while running one to two orders of magnitude faster.

However, the authors highlight a limitation: for matrices whose columns are unit vectors with high mutual coherence—such as Hadamard bases or the discrete cosine transform—every k‑subset of rank r with k<n has a certificate value at least 1/coh(A), which equals √r. In these cases no choice of K can improve the approximation guarantee beyond this factor.

The work combines geometric arguments with classical experimental design theory to establish a deterministic algorithm that delivers sparse, fast-to-evaluate generalized inverses with provable approximation bounds, while also delineating the fundamental limits imposed by column coherence.