Equivariance, Curvature and Symmetry in Functional Covariance Estimation

Abstract: Statistical procedures for functional data are routinely applied after changes of time scale, registration, or other reparametrisations, although it is generally unclear when the resulting inference is independent of the chosen coordinates. We characterize this equivariance for local-linear covariance estimation from sparsely observed functional data. At the population level, covariance operators are unitarily conjugate under every diffeomorphic reparametrisation. At the estimation level, exact commutation holds universally if and only if the reparametrisation is affine. For a general (C^{2,1}) diffeomorphism, departure from equivariance is controlled by the normalized curvature (κψ=|ψ''/ψ'|\infty), with local-linear defect [ O_P!\left{κψ\left(h^2+h n{\mathrm{loc}}^{-1/2} +h_0^2+h_0 n_{\mathrm{loc},1}^{-1/2}\right)\right}. ] Thus zero curvature is exactly the boundary of statistical equivariance. We then show that finite-group symmetry acts as an orthogonal-projection regularizer: its risk gain is exactly the anti-invariant estimation error minus the squared symmetry misspecification. An orbit-covariance identity quantifies the attainable variance reduction and shows why group size alone does not determine the gain. These principles propagate to eigenvalues, eigenspaces and truncated PACE prediction. The results separate coordinate invariance at the population level from the geometric obstructions introduced by statistical smoothing.