Modern satellite-based analysis of the ice sheets presents a profound statistical-geometric problem: how do we make sense of scattered, noisy measurements of vast, steadily evolving surfaces like the Greenland and Antarctic ice sheets? In these lecture notes, I attempt to provide the mathematical foundations of function approximation techniques that may aid the reader in appreciating and tackling this problem. The main topics include non-parametric regression, linear models, Gaussian processes, and reproducing Kernel Hilbert spaces. Each chapter features both examples from recent glaciology research and mathematical “curios” which invoke more niche remarks, such as the duality of Voronoi and Delaunay graphs and the intimate relationship between free knot linear splines and ReLu neural networks. We also have code demos for some of these topics, available on our GitHub repository.
Noah Bergam. (2023, September 20). Regression on Ice: Function approximation for the mathematically-inclined glaciologist. Zenodo. https://doi.org/10.5281/zenodo.8363976
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