Multiscale scattered data analysis in samplet coordinates
This work addresses efficient interpolation for large-scale scattered data in fields like geostatistics or machine learning, but it is incremental as it builds on existing multiscale and samplet techniques.
The paper tackles the problem of multiscale scattered data interpolation for radial basis functions, particularly the Matérn class, by proposing a method that uses samplet coordinates to achieve sparse approximations and bounded condition numbers, resulting in an overall computational cost of O(N log^2 N) for large datasets.
We study multiscale scattered data interpolation schemes for globally supported radial basis functions with focus on the Matérn class. The multiscale approximation is constructed through a sequence of residual corrections, where radial basis functions with different lengthscale parameters are combined to capture varying levels of detail. We prove that the condition numbers of the the diagonal blocks of the corresponding multiscale system remain bounded independently of the particular level, allowing us to use an iterative solver with a bounded number of iterations for the numerical solution. Employing an appropriate diagonal scaling, the multiscale system becomes well conditioned. We exploit this fact to derive a general error estimate bounding the consistency error issuing from a numerical approximation of the multiscale system. To apply the multiscale approach to large data sets, we suggest to represent each level of the multiscale system in samplet coordinates. Samplets are localized, discrete signed measures exhibiting vanishing moments and allow for the sparse approximation of generalized Vandermonde matrices issuing from a vast class of radial basis functions. Given a quasi-uniform set of $N$ data sites, and local approximation spaces with exponentially decreasing dimension, the samplet compressed multiscale system can be assembled with cost $\mathcal{O}(N \log^2 N)$. The overall cost of the proposed approach is $\mathcal{O}(N \log^2 N)$. The theoretical findings are accompanied by extensive numerical studies in two and three spatial dimensions.