## Abstract

It has previously been reported that the representation that is learned in the first layer of deep Convolutional Neural Networks (CNNs) is highly consistent across initializations and architectures. In this work, we quantify this consistency by considering the first layer as a filter bank and measuring its energy distribution. We find that the energy distribution is very different from that of the initial weights and is remarkably consistent across random initializations, datasets, architectures and even when the CNNs are trained with random labels. In order to explain this consistency, we derive an analytical formula for the energy profile of linear CNNs and show that this profile is mostly dictated by the second order statistics of image patches in the training set and it will approach a whitening transformation when the number of iterations goes to infinity. Finally, we show that this formula for linear CNNs also gives an excellent fit for the energy profiles learned by commonly used nonlinear CNNs such as ResNet and VGG, and that the first layer of these CNNs indeed performs approximate whitening of their inputs.

Original language | American English |
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Pages (from-to) | 6115-6139 |

Number of pages | 25 |

Journal | Proceedings of Machine Learning Research |

Volume | 202 |

State | Published - 2023 |

Event | 40th International Conference on Machine Learning, ICML 2023 - Honolulu, United States Duration: 23 Jul 2023 → 29 Jul 2023 |

### Bibliographical note

Publisher Copyright:© 2023 Proceedings of Machine Learning Research. All rights reserved.