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 C2,1 diffeomorphism, departure from equivariance is controlled by the normalized curvature κψ=|ψ''/ψ'|∞, with local-linear defect OP {κψ(h2+h nloc−1/2 +h02+h0 nloc,1−1/2)}. 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.
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BibTeX
@misc{nembe2026equivariance,
title = {{Equivariance, Curvature and Symmetry in Functional Covariance Estimation}},
author = {Jocelyn Nembe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03042},
eprint = {2609.03042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03042},
}
RIS
TY - UNPB
TI - Equivariance, Curvature and Symmetry in Functional Covariance Estimation
AU - Jocelyn Nembe
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.03042
UR - https://arxiv.org/abs/2609.03042
AB - 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 κ_ψ=|ψ''/ψ'|_∞, with local-linear defect O_P {κ_ψ(h^2+h n_loc^−1/2 +h_0^2+h_0 n_loc,1^−1/2)}. 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.
ER -