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Spatially adaptive sparse grid regression
Keywords: Sparse Grids; Spatial Adaptivity; Regression
Mon, 09:00--09:25
  • Pflüger, Dirk (Department of Informatics, Technische Universit√§t M√ľnchen, Germany)

Regression, (high-dimensional) function reconstruction from scattered data, is a common problem in data-driven tasks. Typically, meshfree methods are employed to circumvent the curse of dimensionality. To deal with regression by discretizing the feature space, the sparse grid combination technique has been successfully employed [1]. Due to their primarily data-independent approach, sparse grids enable one to deal with massive amounts of data.

Meanwhile, the direct sparse grid approach has also been employed to a range of data-driven problems such as classification [2]. It allows to adapt to the problem at hand in a spatially adaptive way, which is crucial for many problems. We now present spatially adaptive sparse grids for regression problems. Using a suitable (problem-aware) choice of basis functions, they provide straightforward criteria for adaptive refinement, built-in dimensional adaptivity, and an ANOVA-style decomposition of the problem at hand. Where the effective dimensionality of the problem is moderate, high dimensionalities as well as large data sets can be dealt with. This is confirmed by results for both artificial datasets and real-world ones (e.g., from astrophysics and finance).

[1] Jochen Garcke and Markus Hegland. Fitting multidimensional data using gradient penalties and the sparse grid combination technique. Computing, 84(1-2):1–25, April 2009.
[2] Dirk Pflüger. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Dr. Hut, München, 2010.