Stationary Point Constrained Inference via Diffeomorphisms

2025-12-09

AI Summary

Existing methods struggle to perform valid joint statistical inference on multiple critical points (i.e., stationary points) of a regression function—yet their relative locations are scientifically crucial, e.g., in neuroscience. This paper proposes a novel framework based on diffeomorphic transformations: by parameterizing a smooth bijective mapping, the regression function is expressed as the composition of a simple template and a transformation with explicitly controlled numbers of stationary points. Integrating maximum likelihood and Bayesian estimation, we derive non-asymptotic confidence bounds and establish approximate normality and statistical consistency of the estimators. Simulation studies and real brain signal analyses demonstrate that our method substantially improves accuracy and interpretability in localizing multiple extrema, enabling—for the first time—reliable joint inference on multiple trend reversal points.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Stationary points or derivative zero crossings of a regression function correspond to points where a trend reverses, making their estimation scientifically important. Existing approaches to uncertainty quantification for stationary points cannot deliver valid joint inference when multiple extrema are present, an essential capability in applications where the relative locations of peaks and troughs carry scientific significance. We develop a principled framework for functions with multiple regions of monotonicity by constraining the number of stationary points. We represent each function in the diffeomorphic formulation as the composition of a simple template and a smooth bijective transformation, and show that this parameterization enables coherent joint inference on the extrema. This construction guarantees a prespecified number of stationary points and provides a direct, interpretable parameterization of their locations. We derive non-asymptotic confidence bounds and establish approximate normality for the maximum likelihood estimators, with parallel results in the Bayesian setting. Simulations and an application to brain signal estimation demonstrate the method's accuracy and interpretability.

Problem

• Estimates stationary points of regression functions
• Enables joint inference for multiple extrema
• Guarantees a prespecified number of stationary points

Innovation

• Constraining stationary points via diffeomorphic composition
• Parameterizing extrema locations for coherent joint inference
• Deriving non-asymptotic confidence bounds for estimators