The Universal Approximation Property Characterization, Construction, Representation, and Existence
The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models’ potential. This paper works t...
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| Published in | Annals of mathematics and artificial intelligence Vol. 89; no. 5-6; pp. 435 - 469 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.06.2021
Springer |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1012-2443 1573-7470 |
| DOI | 10.1007/s10472-020-09723-1 |
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| Abstract | The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models’ potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation functions allow the feed-forward architecture to maintain its universal approximation capabilities when multiple constraints are imposed on its final layers and its remaining layers are only sparsely connected. These include a rescaled and shifted Leaky ReLU activation function but not the ReLU activation function. Our construction and representation result is used to exhibit a simple modification of the feed-forward architecture, which can approximate any continuous function with non-pathological growth, uniformly on the entire Euclidean input space. This improves the known capabilities of the feed-forward architecture. |
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| AbstractList | The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models’ potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation functions allow the feed-forward architecture to maintain its universal approximation capabilities when multiple constraints are imposed on its final layers and its remaining layers are only sparsely connected. These include a rescaled and shifted Leaky ReLU activation function but not the ReLU activation function. Our construction and representation result is used to exhibit a simple modification of the feed-forward architecture, which can approximate any continuous function with non-pathological growth, uniformly on the entire Euclidean input space. This improves the known capabilities of the feed-forward architecture. The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models' potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation functions allow the feed-forward architecture to maintain its universal approximation capabilities when multiple constraints are imposed on its final layers and its remaining layers are only sparsely connected. These include a rescaled and shifted Leaky ReLU activation function but not the ReLU activation function. Our construction and representation result is used to exhibit a simple modification of the feed-forward architecture, which can approximate any continuous function with non-pathological growth, uniformly on the entire Euclidean input space. This improves the known capabilities of the feed-forward architecture. Keywords Universal approximation * Constrained approximation * Uniform approximation * Deep learning * Topological transitivity * Composition operators Mathematics Subject Classification (2010) 68T07 47B33 * 47A16 * 68T05 * 30L05 * 46M40 |
| Audience | Academic |
| Author | Kratsios, Anastasis |
| Author_xml | – sequence: 1 givenname: Anastasis orcidid: 0000-0001-6791-3371 surname: Kratsios fullname: Kratsios, Anastasis email: anastasis.kratsios@math.ethz.ch organization: (ETH) Eidgenössische Technische Hochschule Zürich |
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| Keywords | Deep learning 30L05 46M40 68T05 47A16 Universal approximation Uniform approximation Composition operators Constrained approximation 68T07 47B33 Topological transitivity |
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| SubjectTerms | Artificial Intelligence Complex Systems Computer Science Costs (Law) Machine learning Mathematics Neural networks |
| Subtitle | Characterization, Construction, Representation, and Existence |
| Title | The Universal Approximation Property |
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