Equivariance, Curvature and Symmetry in Functional Covariance Estimation

What the math gives to ML

The paper separates exact population-level coordinate invariance from the non-invariance introduced by finite-bandwidth statistical smoothing. Its transferable asset is a quantitative defect bound: affine reparameterizations commute exactly, while nonlinear warps produce an error proportional to normalized curvature and explicit bandwidth and sparsity terms. This supports curvature-aware consistency training for neural models processing irregular time series or continuous signals. Rather than imposing impossible exact invariance under every warp, the model can enforce a tolerance calibrated to the warp geometry and effective local sample size.