Christoffel Adaptive Sampling for Sparse Random Feature Expansions
This work addresses data efficiency in scientific computing where data acquisition is costly, though it appears incremental as it builds on existing SRFE methods by adding adaptive sampling.
The paper tackled the problem of improving sampling efficiency in Sparse Random Feature Expansions (SRFE) for function approximation by integrating active learning with adaptive sampling guided by the Christoffel function, resulting in maintained high accuracy while reducing sample complexity in data-scarce settings.
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.