A JOHNSON-LINDENSTRAUSS FRAMEWORK FOR RANDOMLY INITIALIZED CNNS

Ido Nachum, Jan Hazła, Michael Gastpar, Anatoly Khina

Research output: Contribution to conferencePaperpeer-review

1 Scopus citations

Abstract

How does the geometric representation of a dataset change after the application of each randomly initialized layer of a neural network? The celebrated Johnson-Lindenstrauss lemma answers this question for linear fully-connected neural networks (FNNs), stating that the geometry is essentially preserved. For FNNs with the ReLU activation, the angle between two inputs contracts according to a known mapping. The question for non-linear convolutional neural networks (CNNs) becomes much more intricate. To answer this question, we introduce a geometric framework. For linear CNNs, we show that the Johnson-Lindenstrauss lemma continues to hold, namely, that the angle between two inputs is preserved. For CNNs with ReLU activation, on the other hand, the behavior is richer: The angle between the outputs contracts, where the level of contraction depends on the nature of the inputs. In particular, after one layer, the geometry of natural images is essentially preserved, whereas for Gaussian correlated inputs, CNNs exhibit the same contracting behavior as FNNs with ReLU activation.

Original languageEnglish
StatePublished - 2022
Event10th International Conference on Learning Representations, ICLR 2022 - Virtual, Online
Duration: 25 Apr 202229 Apr 2022

Conference

Conference10th International Conference on Learning Representations, ICLR 2022
CityVirtual, Online
Period25/04/2229/04/22

Funding

FundersFunder number
Israel Ministry of Economy and Industry
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung200364
Israel Science Foundation2077/20

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