Lexicographic functional calculus and its application to functional calculus calculus
arXiv:2608.08404
2026
Architecture
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper develops lexicographic functional calculus, an ordered noncommutative calculus for differentiating matrix functions with perturbation operators inserted between spectral factors. Its transferable asset is an explicit higher-order Fréchet derivative formula expressed through divided differences and spectral projectors, including the noncommuting case and repeated eigenvalues. This can become a robust spectral neural-network layer with custom Jacobian-vector and Hessian-vector products, avoiding unstable eigenvector differentiation and finite-difference approximations.
Ideas from this paper
Unverified
2026
Replace an ordinary elementwise nonlinearity on a learned Hermitian matrix with a matrix function f(A), while supplying exact Jacobian-vector and Hessian-vector products through the lexicographic divided-difference formula. This gives a principled spectral layer for covariance features, graph operators, attention kernels, or matrix-valued embeddings, particularly when perturbation matrices do not commute and eigenvalues are repeated or nearly repeated.
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