Randomized Block Davidson Eigensolvers for Plane-Wave Density-Functional Theory

What the math gives to ML

The transferable asset is randomized Gram-Schmidt: a large block of vectors can be orthogonalized using a low-dimensional sketched inner product instead of repeated full-dimensional dot products. The resulting basis need not be Euclidean orthonormal, but exact Rayleigh-Ritz extraction is recovered by solving a definite generalized Hermitian eigenproblem with the true overlap matrix. This suggests a curvature-subspace optimizer that builds Hessian or Fisher directions from gradients and Hessian-vector products, uses sketching to avoid expensive basis orthogonalization, and performs exact small-matrix mode extraction before applying a damped update. The method is most promising when parameter dimension is very large and the retained subspace is moderately wide.