Semiparametric Learning of Integral Functionals on Submanifolds

2025-07-16

AI Summary

This paper addresses semiparametric estimation and inference for integral functionals defined on $m$-dimensional submanifolds—objects prevalent in econometrics. To mitigate the “curse of dimensionality” inherent in nonparametric estimation in ambient $d$-dimensional space, we exploit intrinsic low-dimensional geometric structure by integrating over the $m$-dimensional submanifold, effectively reducing estimation complexity from $d$ to $(d-m)$ dimensions. We construct a plug-in semiparametric estimator leveraging Hölder smoothness assumptions and derive its minimax-optimal convergence rate $n^{-s/(2s+d-m)}$. Under both linear and nonlinear settings, we establish asymptotic normality and develop consistent variance estimators. Simulation studies corroborate the theoretical properties. Our framework provides a novel paradigm for inference on high-dimensional structured functions, bridging differential geometry with semiparametric statistics.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

This paper studies the semiparametric estimation and inference of integral functionals on submanifolds, which arise naturally in a variety of econometric settings. For linear integral functionals on a regular submanifold, we show that the semiparametric plug-in estimator attains the minimax-optimal convergence rate $n^{-frac{s}{2s+d-m}}$, where $s$ is the Hölder smoothness order of the underlying nonparametric function, $d$ is the dimension of the first-stage nonparametric estimation, $m$ is the dimension of the submanifold over which the integral is taken. This rate coincides with the standard minimax-optimal rate for a $(d-m)$-dimensional nonparametric estimation problem, illustrating that integration over the $m$-dimensional manifold effectively reduces the problem's dimensionality. We then provide a general asymptotic normality theorem for linear/nonlinear submanifold integrals, along with a consistent variance estimator. We provide simulation evidence in support of our theoretical results.

Problem

• Estimating integral functionals on submanifolds semiparametrically
• Determining minimax-optimal convergence rates for estimators
• Proving asymptotic normality for submanifold integral estimators

Innovation

• Semiparametric plug-in estimator for integral functionals
• Minimax-optimal convergence rate analysis
• Asymptotic normality theorem for submanifold integrals