Rigorous theoretical justification of perturbation-based comparability for CUR error bounds

Establish a rigorous theoretical justification that, for a matrix A perturbed to B = A + tE where E is totally positive and t > 0 is sufficiently small, the subsets (S, W) obtained by applying Algorithm 1 to B yield an approximation for A such that, whenever A_{S,W} is invertible, the spectral norm error ||A − A_{:,W}(A_{S,W})^{-1}A_{S,:}||_2^2 is comparable to ||B − B_{:,W}(B_{S,W})^{-1}B_{S,:}||_2^2; derive explicit bounds quantifying this comparability and specify assumptions on t and E under which the comparison holds.

Background

The deterministic algorithm proposed for classical CUR requires the assumption that no square submatrix of A of size at most k is singular. To bypass this limitation in practice, the authors suggest perturbing A to B = A + tE with a small totally positive matrix E so B lacks such singular submatrices, and then running Algorithm 1 on B to obtain S and W.

They observe empirically that if A_{S,W} is invertible, the CUR approximation error for A appears comparable to that for B and thus effectively bounded. However, a rigorous theoretical explanation and quantified guarantees for this observation are not provided, motivating the open problem.

References

Numerical experiments suggest that if A_{\widehat{S}, \widehat{W}} is invertible, then the approximation error \Vert A−A_{:,\widehat{W}}(A_{\widehat{S},\widehat{W}}){-1}A_{\widehat{S},:}\Vert_{2}{2} is comparable to \Vert B−B_{:,\widehat{W}}(B_{\widehat{S},\widehat{W}}){-1}B_{\widehat{S},:}\Vert_{2}{2, and can therefore also be effectively bounded. A rigorous theoretical understanding of this observation is left for future work.

— Generalized Interlacing Families: New Error Bounds for CUR Matrix Decompositions (2512.07903 - Cai et al., 7 Dec 2025) in Remark after Theorem th3 in the subsection “Classical CUR matrix decompositions”