Journal article
Approximation Properties of Ridge Functions and Extreme Learning Machines
SIAM journal on mathematics of data science, Vol.3(3), pp.815-832
01/01/2021
DOI: 10.1137/20M1356348
Abstract
For a compact set D subset of R-m we consider the problem of approximating a function f over D by sums of ridge functions x bar right arrow phi(w(T)x) with w in a given set W. While such sums are dense in C(D) if W = R-m, to understand the effectiveness of these approximations we consider finite W and estimate the infimum of parallel to f - g parallel to(infinity) over g is an element of V-w := span { x bar right arrow phi(w(T)x) phi is an element of C (R), w is an element of W } in terms of the Lipschitz constant of f. In particular, we show lower bounds for the worst case approximation of functions of Lipschitz constant one by considering unapproximable functions f (parallel to f - g parallel to(infinity) >= parallel to f parallel to(infinity)- for any g is an element of V-w) of Lipschitz constant one. Accurate approximations appear to require vertical bar W vertical bar = Omega(m(2)) for D a unit hypercube in R-m: if vertical bar W vertical bar <= 1/2m(m - 1) and m >= 2, then sup(f) inf(g)is an element of V-w parallel to f - g parallel to(infinity), >= 1/ root m- 1, where f ranges over Lipschitz functions of Lipschitz constant one. Similar results hold for sums of products of pairs of ridge functions with weight vectors in W.
Details
- Title: Subtitle
- Approximation Properties of Ridge Functions and Extreme Learning Machines
- Creators
- Palle Jorgensen - Univ Iowa, Dept Math, Iowa City, IA 52242 USADavid E Stewart - Univ Iowa, Dept Math, Iowa City, IA 52242 USA
- Resource Type
- Journal article
- Publication Details
- SIAM journal on mathematics of data science, Vol.3(3), pp.815-832
- Publisher
- SIAM PUBLICATIONS
- DOI
- 10.1137/20M1356348
- ISSN
- 2577-0187
- eISSN
- 2577-0187
- Number of pages
- 18
- Language
- English
- Date published
- 01/01/2021
- Academic Unit
- Mathematics
- Record Identifier
- 9984242443902771
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