A note on the convergence analysis of a sparse grid multivariate probability density estimator

Abstract

With the recent growth in volume and complexity of available data has come a renewed interest in the problem of estimating multivariate probability density functions. However, traditional methods encounter the curse of dimensionality (complexity grows exponentially with dimension). Here we provide an outline of a convergence analysis of a sparse grid based probability density estimation, which supports the use of the method for moderately complex (up to 15 dimensions) data sets, as has already been demonstrated for sparse grid quadrature and interpolation. References Hans-Joachim Bungartz and Michael Griebel. A note on the complexity of solving {P}oisson's equation for spaces of bounded mixed derivatives. J. Complexity, 15(2):167--199, 1999. http://dx.doi.org/10.1006/jcom.1999.0499 Hans-Joachim Bungartz and Michael Griebel. Sparse grids. Acta Numerica, 13:147--269, 2004. http://dx.doi.org/10.1017/S0962492904000182 J. Garcke. Sparse grid tutorial. http://www.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf [29 January, 2009], 2008. J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing, 67:225--253, 2001. http://dx.doi.org/10.1007/s006070170007 D. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York, 1992. C. Zenger. Sparse grids. In W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations: Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, volume 31 of Notes on Num. Fluid Mech., pages 241--251. Vieweg, 1991.