%0 Journal Article %1 zhang2019approximation %A Zhang, Han %A Gao, Xi %A Unterman, Jacob %A Arodz, Tom %D 2019 %K approximate graphs inverse physics readings %T Approximation Capabilities of Neural ODEs and Invertible Residual Networks %U http://arxiv.org/abs/1907.12998 %X Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far the question of whether Neural ODEs and i-ResNets can model any continuous invertible function remained unresolved. Here, we show that both of these models are limited in their approximation capabilities. We then prove that any homeomorphism on a $p$-dimensional Euclidean space can be approximated by a Neural ODE operating on a $2p$-dimensional Euclidean space, and a similar result for i-ResNets. We conclude by showing that capping a Neural ODE or an i-ResNet with a single linear layer is sufficient to turn the model into a universal approximator for non-invertible continuous functions.

@article{zhang2019approximation, abstract = {Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far the question of whether Neural ODEs and i-ResNets can model any continuous invertible function remained unresolved. Here, we show that both of these models are limited in their approximation capabilities. We then prove that any homeomorphism on a $p$-dimensional Euclidean space can be approximated by a Neural ODE operating on a $2p$-dimensional Euclidean space, and a similar result for i-ResNets. We conclude by showing that capping a Neural ODE or an i-ResNet with a single linear layer is sufficient to turn the model into a universal approximator for non-invertible continuous functions.}, added-at = {2020-06-04T21:35:15.000+0200}, author = {Zhang, Han and Gao, Xi and Unterman, Jacob and Arodz, Tom}, biburl = {https://www.bibsonomy.org/bibtex/27c6edc7be6a39e09bf0148cdc87fea98/kirk86}, description = {[1907.12998] Approximation Capabilities of Neural ODEs and Invertible Residual Networks}, interhash = {4538f7abe699856c6091d9cb59316942}, intrahash = {7c6edc7be6a39e09bf0148cdc87fea98}, keywords = {approximate graphs inverse physics readings}, note = {cite arxiv:1907.12998}, timestamp = {2020-06-04T21:35:15.000+0200}, title = {Approximation Capabilities of Neural ODEs and Invertible Residual Networks}, url = {http://arxiv.org/abs/1907.12998}, year = 2019 }