Machine Learning Summer School 2006 - Canberra

The Sparse Grid Method

author: Jochen Garcke, Australian National University - ANU

Description

The sparse grid method is a special discretization technique, which allows to cope with the curse of dimensionality to some extent. It is based on a hierarchical basis and a sparse tensor product decompositon. Sparse grids have been successfully used to solve partial differential equations in the past and, more recently, have been shown to be competitive for learning problems as well. The lecture will provide a general introduction to the major properties of sparse grids and present the sparse grid combination technique for classification and regression.

Categories

Top: Computer Science: Machine Learning: Preprocessing

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*Slides*
0:00	Sparse Grids
0:02	Outline
0:31	Partial Differential Equations
5:05	Galerkin-Variational Principle
7:30	Discretisation
9:01	Example for VN in One Dimension
10:04	One-dimensional Basis Functions
10:43	Basis Functions in More Dimensions
12:40	Some Notation
14:14	Triangulation Instead of Tensor Product
15:47	Approximation Properties
20:52	Interpolation with Hierarchical Basis
23:31	Hierarchical Difference Spaces
24:43	Hierarchical Tensor Product Decomposition
25:33	Hierarchical Subspaces Wl for V3,3
26:11	Hierarchical Basis [Faber:09,Yserentant:86]
27:00	Interpolation with Hierarchical Basis
28:33	Sobolev-Space H2 mix with Domin. Mixed Deriv.
30:32	Hierarchical Values l;j are Bounded I
31:24	Hierarchical Values l;j are Bounded II
33:26	Hierarchical Values l;j are Bounded III
34:26	Hier. Compon. Bounded by Size of Support
35:48	Hierarchical Subspaces Wl
36:11	Hierarchical Subspaces Wl 01
36:43	Sparse Grids
38:56	Sparse Grids in two and three dimensions
39:24	History of Sparse Grids
40:37	Some Recent Applications of Sparse Grids
40:54	Simple Example in Numerical Integration 10D
43:11	How to Compute on a Sparse Grid
43:54	Combination Technique of Level 4 in 2d
44:59	Telescope Sum Property for Interpolation
45:15	Sparse Grid Combination Technique
47:16	Generalised Combination Technique
48:23	Summary Sparse Grids
49:11	Problem Setting for Regression / Classfication
50:02	Regularisation Theory
51:25	Discretisation

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