Towards an Asymptotic Efficiency Theory on Regular Parameter Manifolds

2025-10-15

AI Summary

Classical asymptotic efficiency theory assumes sample and parameter spaces are normed linear spaces, rendering it inadequate for complex manifold-structured data. Method: This project establishes a unified asymptotic efficiency theory on Riemannian manifolds, systematically extending classical efficiency analysis to settings where both sample and parameter spaces are regular Riemannian manifolds. It introduces core concepts—including regular estimators and differentiable functionals on manifolds—and integrates Riemannian geometry, influence functions, and differential statistical methods to develop a technical framework for asymptotic unbiasedness, local regularity, and Cramér–Rao-type efficiency lower bounds on manifolds. Contribution/Results: The theory is validated in Fréchet mean estimation and single-index model regression coefficient estimation, yielding tight efficiency bounds and substantially enhancing statistical inference capability and interpretability for non-Euclidean structured data.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Asymptotic efficiency theory is one of the pillars in the foundations of modern mathematical statistics. Not only does it serve as a rigorous theoretical benchmark for evaluating statistical methods, but it also sheds light on how to develop and unify novel statistical procedures. For example, the calculus of influence functions has led to many important statistical breakthroughs in the past decades. Responding to the pressing challenge of analyzing increasingly complex datasets, particularly those with non-Euclidean/nonlinear structures, many novel statistical models and methods have been proposed in recent years. However, the existing efficiency theory is not always readily applicable to these cases, as the theory was developed, for the most part, under the often neglected premise that both the sample space and the parameter space are normed linear spaces. As a consequence, efficiency results outside normed linear spaces are quite rare and isolated, obtained on a case-by-case basis. This paper aims to develop a more unified asymptotic efficiency theory, allowing the sample space, the parameter space, or both to be Riemannian manifolds satisfying certain regularity conditions. We build a vocabulary that helps translate essential concepts in efficiency theory from normed linear spaces to Riemannian manifolds, such as (locally) regular estimators, differentiable functionals, etc. Efficiency bounds are established under conditions parallel to those for normed linear spaces. We also demonstrate the conceptual advantage of the new framework by applying it to two concrete examples in statistics: the population Frechet mean and the regression coefficient vector of Single-Index Models.

Problem

• Extends asymptotic efficiency theory to Riemannian manifolds
• Addresses limitations of existing theory for nonlinear structures
• Provides efficiency bounds for statistical models on manifolds

Innovation

• Extends efficiency theory to Riemannian manifolds
• Translates statistical concepts to curved spaces
• Establishes efficiency bounds for non-Euclidean data