Local and global topological complexity measures OF ReLU neural network functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916188037054464 |
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| author | Grigsby, J. Elisenda Lindsey, Kathryn Masden, Marissa |
| author_facet | Grigsby, J. Elisenda Lindsey, Kathryn Masden, Marissa |
| contents | We apply a generalized piecewise-linear (PL) version of Morse theory due to Grunert-Kuhnel-Rote to define and study new local and global notions of topological complexity for fully-connected feedforward ReLU neural network functions, F: R^n -> R. Along the way, we show how to construct, for each such F, a canonical polytopal complex K(F) and a deformation retract of the domain onto K(F), yielding a convenient compact model for performing calculations. We also give a construction showing that local complexity can be arbitrarily high. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_06062 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Local and global topological complexity measures OF ReLU neural network functions Grigsby, J. Elisenda Lindsey, Kathryn Masden, Marissa Algebraic Topology Computational Geometry Machine Learning Geometric Topology 57R70, 57Q99, 52B70, 52C35 We apply a generalized piecewise-linear (PL) version of Morse theory due to Grunert-Kuhnel-Rote to define and study new local and global notions of topological complexity for fully-connected feedforward ReLU neural network functions, F: R^n -> R. Along the way, we show how to construct, for each such F, a canonical polytopal complex K(F) and a deformation retract of the domain onto K(F), yielding a convenient compact model for performing calculations. We also give a construction showing that local complexity can be arbitrarily high. |
| title | Local and global topological complexity measures OF ReLU neural network functions |
| topic | Algebraic Topology Computational Geometry Machine Learning Geometric Topology 57R70, 57Q99, 52B70, 52C35 |
| url | https://arxiv.org/abs/2204.06062 |