arXiv subjects

Jocelyn Nembe

Publications and source records attributed to Jocelyn Nembe.

3 recordsLinked to original sources

arXiv2026

Equivariance, Curvature and Symmetry in Functional Covariance Estimation

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 $κ_ψ=\|ψ''/ψ'\|_\infty$, with local-linear defect \[ O_P\!\left\{κ_ψ\left(h^2+h n_{\mathrm{loc}}^{-1/2} +h_0^2+h_0 n_{\mathrm{loc},1}^{-1/2}\right)\right\}. \] 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.

math.ST↗

arXiv2026

How far can symmetry help? Phase transitions and symmetry selection in sparse functional data analysis

In sparse functional data analysis, where $n$ curves are each observed at $m$ random points, the covariance surface undergoes a sharp phase transition: if the covariance has smoothness $β$, the risk drops from the two-dimensional nonparametric rate to the parametric rate $n^{-1}$ once $m$ exceeds $m^*_n \asymp n^{1/(2β)}$. We determine what a symmetry of the domain does to that transition. A cyclic group of order $q$ preserving process and design displaces the threshold to $n^{1/(2β)}q^{-1/2}$; the exponent is a square root because symmetry acts on the number of usable pairs, which enters the variance quadratically, while the parametric floor is untouched by group averaging. The displacement saturates: once the orbit is finer than the bandwidth the reduction factor is $\min(q,c_K/h)$, by Poisson summation and positive definiteness of the kernel autocorrelation, and beyond that point the rate collapses to the one-dimensional nonparametric rate. Hence no rotation symmetry, even the full circle group, lowers the threshold below $n^{1/(4β)}$; this floor is attained from above by our estimator and from below, up to a polynomial factor, by our lower bounds. Symmetry thus takes one halfway on a logarithmic scale from the classical threshold to constant sampling. The uniform lower bound rests on a positivity-preserving packing of the stationary sub-class; closing the gap is left open. If the symmetry is only approximate, the risk acquires an approximation term and the design plane splits into three regimes, one unreachable by additional sampling; the symmetry spectrum governing it is explicit in Fourier coordinates, and a hold-out procedure selects the symmetry level with leading constant one below saturation and within an absolute constant beyond. All laws above are confirmed numerically. Reparametrisation of the domain, by contrast, leaves the threshold unchanged.

math.ST↗

arXiv2025

Adaptive Nonparametric Estimation via Kernel Transport on Group Orbits: Oracle Inequalities and Minimax Rates

We develop a unified framework for nonparametric functional estimation based on kernel transport along orbits of discrete group actions, which we term \emph{Twin Spaces}. Given a base kernel $K$ and a group $G = \langleφ\rangle$ acting isometrically on the input space $E$, we construct a hierarchy of transported kernels $\{K_j\}_{j\geq 0}$ and a penalized model selection scheme satisfying a Kraft inequality. Our main contributions are threefold: (i) we establish non-asymptotic oracle inequalities for the penalized twin-kernel estimator with explicit constants; (ii) we introduce novel twin-regularity classes that capture smoothness along group orbits and prove that our estimator adapts to these classes; (iii) we show that the framework recovers classical minimax-optimal rates in the Euclidean setting while enabling improved rates when the target function exhibits orbital structure. The effective dimension $d_{\mathrm{eff}}$ governing the rates is characterized in terms of the quotient $G/L$, where $L$ is the subgroup preserving the base operation. Connections to wavelet methods, geometric quantization, and adaptive computation are discussed.

math.ST↗