Book chapter
Ridge Function Machines
Explorations in the Mathematics of Data Science, pp.85-100
Applied and Numerical Harmonic Analysis, Springer Nature Switzerland
2024
DOI: 10.1007/978-3-031-66497-7_5
Abstract
Ridge function machines implement a computationally tractable and theoretically sound means of creating approximations to functions given by data. A ridge function is a function .Rn → R given by .x | → g(wT x) where .w ∈ Rn and .g : R → R. A ridge function machine is a sum of ridge functions . ∑m j =1 g j (wT j x). While it is computationally infeasible to allow for arbitrary continuous functions .g j in this sum, we can approximate such functions by means of linear combinations of B-splines, for example. Given data points .(xi , yi ), .i = 1, 2 , . . . , m, the regularized ridge function approximation problem .ming∈G N−1 ∑N i=1(g(xi ) − yi )2 + α ‖g‖2 L2 can be solved in .O(N +p) time where .G is the span of p uniformly spaced B-splines. For approximation by sums of ridge functions, a block Gauss–Seidel method can be used. The use of a limited number of weight vectors .wj means that the method effectively performs an orthogonal projection onto a low or modest-dimensional subspace, an example of dimension reduction. Gradients with respect to the weight vectors .wj can also be efficiently computed.
Details
- Title: Subtitle
- Ridge Function Machines
- Creators
- David E. Stewart
- Contributors
- Simon Foucart (Editor)Stephan Wojtowytsch (Editor)
- Resource Type
- Book chapter
- Publication Details
- Explorations in the Mathematics of Data Science, pp.85-100
- Publisher
- Springer Nature Switzerland; Cham
- Series
- Applied and Numerical Harmonic Analysis
- DOI
- 10.1007/978-3-031-66497-7_5
- eISSN
- 2296-5017
- ISSN
- 2296-5009
- Number of pages
- 16
- Language
- English
- Date published
- 2024
- Academic Unit
- Mathematics
- Record Identifier
- 9984721145802771
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