Approximating matrix functions by block Krylov methods with randomized vectors

What the math gives to ML

The paper develops randomized block Krylov approximations for applying a matrix function to a vector without forming the full matrix function, using repeated matrix-vector or matrix-block products and a small projected matrix. The transferable asset is that a few random probe directions can enrich the Krylov subspace and reduce the number of expensive operator applications needed for a target approximation. A direct neural-network use is a graph diffusion or spectral filter layer, where a sparse graph operator is repeatedly applied to node features; augmenting the feature block with random probes may achieve the same diffusion accuracy with fewer propagation steps. This is most promising when the graph operator is sparse, the feature dimension is moderate, and matrix-function evaluation is currently implemented by many sequential message-passing steps.