Christoffel Adaptive Sampling for Sparse Random Feature Expansions

TL;DR

This paper introduces a Christoffel function-based adaptive sampling method for Sparse Random Feature Expansions, significantly improving sampling efficiency and accuracy in function approximation tasks with limited data.

Contribution

It presents a novel adaptive sampling approach integrating the Christoffel function into SRFE, enhancing sample efficiency and approximation accuracy over traditional methods.

Findings

Adaptive sampling outperforms nonadaptive in accuracy

Reduces sample complexity in function approximation

Effective in data-scarce scientific computing tasks

Abstract

Random Feature Models (RFMs) have become a powerful tool for approximating multivariate functions and solving partial differential equations efficiently. Sparse Random Feature Expansions (SRFE) improve traditional RFMs by incorporating sparsity, making it particularly effective in data-scarce settings. In this work, we integrate active learning with sparse random feature approximations to improve sampling efficiency. Specifically, we incorporate the Christoffel function to guide an adaptive sampling process, dynamically selecting informative sample points based on their contribution to the function space. This approach optimizes the distribution of sample points by leveraging the Christoffel function associated with an iteratively-chosen basis obtained by the sparse recovery solver. We conduct numerical experiments comparing adaptive and nonadaptive sampling strategies with the SRFE framework and examine their accuracy for various function approximation tasks. Overall, our results demonstrate the advantages of adaptive sampling in maintaining high accuracy while reducing sample complexity for SRFE, highlighting its potential for scientific computing tasks where data is expensive to acquire.