## Abstract

Assessing the complexity of functions computed by a neural network helps us understand how the network will learn and generalize. One natural measure of complexity is how the network distorts length - if the network takes a unit-length curve as input, what is the length of the resulting curve of outputs? It has been widely believed that this length grows exponentially in network depth. We prove that in fact this is not the case: the expected length distortion does not grow with depth, and indeed shrinks slightly, for ReLU networks with standard random initialization. We also generalize this result by proving upper bounds both for higher moments of the length distortion and for the distortion of higher-dimensional volumes. These theoretical results are corroborated by our experiments.

Original language | English (US) |
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State | Published - 2022 |

Event | 10th International Conference on Learning Representations, ICLR 2022 - Virtual, Online Duration: Apr 25 2022 → Apr 29 2022 |

### Conference

Conference | 10th International Conference on Learning Representations, ICLR 2022 |
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City | Virtual, Online |

Period | 4/25/22 → 4/29/22 |

## All Science Journal Classification (ASJC) codes

- Language and Linguistics
- Computer Science Applications
- Education
- Linguistics and Language