Optimal near-optimality bounds for the Lanczos method for matrix functions

What the math gives to ML

The paper proves a sharp near-optimality guarantee for Lanczos approximations of matrix functions applied to vectors, measured in a matrix-power norm. The transferable asset is an explicit comparison between a short Krylov computation and the best possible approximation in the same Krylov subspace, with degradation controlled by the condition number of the positive-definite operator. This can replace expensive eigendecompositions or repeated matrix-function operations inside graph neural networks, whitening layers, implicit solvers, and diffusion modules. The most practical targets are inverse and inverse-square-root functions, where matrix-free sparse matrix-vector products make the method substantially cheaper than dense spectral computation.