arXiv subjects

Jocelyn Nembé

Publications and source records attributed to Jocelyn Nembé.

3 recordsLinked to original sources

arXiv2025

Spectral Equivariance and Geometric Transport in Reproducing Kernel Hilbert Spaces: A Unified Framework for Orthogonal Polynomial and Kernel Estimation

We develop a unified geometric framework for nonparametric estimation based on the notion of Twin Kernel Spaces, defined as orbits of a reproducing kernel under a group action. This structure induces a family of transported RKHS geometries in which classical orthogonal polynomial estimators, kernel estimators, and spectral smoothing methods arise as projections onto transported eigenfunction systems. Our main contribution is a Spectral Equivariance Theorem showing that the eigenfunctions of any transported kernel are obtained by unitary transport of the base eigenfunctions. As a consequence, orthogonal polynomial estimators are equivariant under geometric deformation, kernel estimators correspond to soft spectral filtering in a twin space, and minimax rates and bias--variance tradeoffs are invariant under transport. We provide examples based on Hermite and Legendre polynomials, affine and Gaussian groups, and illustrate the effectiveness of twin transport for adaptive and multimodal estimation. The framework reveals a deep connection between group actions, RKHS geometry, and spectral nonparametrics, offering a unified perspective that encompasses kernel smoothing, orthogonal series, splines, and multiscale methods.

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arXiv2025

TwinKernel Estimation for Point Process Intensity Functions: Adaptive Nonparametric Methods via Orbital Regularity

We develop TwinKernel methods for nonparametric estimation of intensity functions of point processes. Building on the general TwinKernel framework and combining it with martingale techniques for counting processes, we construct estimators that adapt to orbital regularity of the intensity function. Given a point process $N$ with intensity $λ$ and a cyclic group $G = \langleφ\rangle$ acting on the time/space domain, we transport kernels along group orbits to create a hierarchy of smoothed Nelson-Aalen type estimators. Our main results establish: (i) uniform consistency via martingale concentration inequalities; (ii) optimal convergence rates for intensities in twin-Hölder classes, with rates depending on the effective dimension $d_{\mathrm{eff}}$; (iii) adaptation to unknown smoothness through penalized model selection; (iv) automatic boundary bias correction via local polynomial extensions in twin coordinates; (v) minimax lower bounds showing rate optimality. We apply the methodology to hazard rate estimation under random censoring, where periodicity or other orbital structure in the hazard may arise from circadian rhythms, seasonal effects, or treatment schedules. Martingale central limit theorems yield asymptotic confidence bands. Simulation studies demonstrate 3--7$\times$ improvements over classical kernel hazard estimators when the intensity exhibits orbital regularity.

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arXiv2025

Concentration of Measure under Diffeomorphism Groups: A Universal Framework with Optimal Coordinate Selection

We establish a universal framework for concentration inequalities based on invariance under diffeomorphism groups. Given a probability measure $μ$ on a space $E$ and a diffeomorphism $ψ: E \to F$, concentration properties transfer covariantly: if the pushforward $ψ_*μ$ concentrates, so does $μ$ in the pullback geometry. This reveals that classical concentration inequalities -- Hoeffding, Bernstein, Talagrand, Gaussian isoperimetry -- are manifestations of a single principle of \emph{geometric invariance}. The choice of coordinate system $ψ$ becomes a free parameter that can be optimized. We prove that for any distribution class $\Pc$, there exists an optimal diffeomorphism $ψ^*$ minimizing the concentration constant, and we characterize $ψ^*$ in terms of the Fisher-Rao geometry of $\Pc$. We establish \emph{strict improvement theorems}: for heavy-tailed or multiplicative data, the optimal $ψ$ yields exponentially tighter bounds than the identity. We develop the full theory including transportation-cost inequalities, isoperimetric profiles, and functional inequalities, all parametrized by the diffeomorphism group $\Diff(E)$. Connections to information geometry (Amari's $α$-connections), optimal transport with general costs, and Riemannian concentration are established. Applications to robust statistics, multiplicative models, and high-dimensional inference demonstrate that coordinate optimization can improve statistical efficiency by orders of magnitude.

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