Nonparametric estimation of homogenized invariant measures from multiscale data via Hermite expansion

2025-10-29

AI Summary

This paper addresses the nonparametric estimation of the stationary invariant density of homogenized models for multiscale Langevin diffusions. Given continuous-time trajectory data from the full system, we propose a spectral estimator based on the Hermite function basis: truncated Fourier coefficients are computed directly from trajectories via the ergodic theorem, and the invariant density is reconstructed accordingly. The method requires neither explicit solution of the effective dynamics nor prior estimation of the scale-separation parameter, ensuring both theoretical interpretability and computational feasibility. We establish uniform convergence of the estimator to the target invariant density as the scale-separation parameter tends to zero; numerical experiments confirm its robustness and high accuracy across varying degrees of scale separation. The key innovation lies in the first systematic application of Hermite spectral methods to learning invariant densities of multiscale diffusions, enabling end-to-end, model-free density estimation—from full-system trajectories to statistical quantities of the homogenized model.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

We consider the problem of density estimation in the context of multiscale Langevin diffusion processes, where a single-scale homogenized surrogate model can be derived. In particular, our aim is to learn the density of the invariant measure of the homogenized dynamics from a continuous-time trajectory generated by the full multiscale system. We propose a spectral method based on a truncated Fourier expansion with Hermite functions as orthonormal basis. The Fourier coefficients are computed directly from the data owing to the ergodic theorem. We prove that the resulting density estimator is robust and converges to the invariant density of the homogenized model as the scale separation parameter vanishes, provided the time horizon and the number of Fourier modes are suitably chosen in relation to the multiscale parameter. The accuracy and reliability of this methodology is further demonstrated through a series of numerical experiments.

Problem

• Estimating homogenized invariant measures from multiscale data
• Learning invariant density of homogenized dynamics
• Developing spectral density estimation via Hermite expansion

Innovation

• Hermite spectral method for multiscale density estimation
• Fourier coefficients computed via ergodic theorem
• Robust estimator converging to homogenized invariant density