Stable Recovery of Matrix Gauge Classes from Pointwise Invariants

What the math gives to ML

The paper identifies a failure mode of supervising matrix-valued neural outputs through pointwise spectra: eigenvalues do not determine a family of matrices up to one shared orthogonal gauge. Its transferable asset is the construction of gauge-invariant relational features, especially products around pairs or cycles, together with evaluation modulo a single global orthogonal transformation. A practical neural-network adaptation is to train matrix outputs with a spectrum loss augmented by trace products of outputs at several inputs, while validating with orthogonal Procrustes alignment. This is relevant wherever latent channels, orbitals, or state bases are physically interchangeable.