A Johnson--Lindenstrauss Framework for Randomly Initialized CNNs

TL;DR

This paper extends the Johnson--Lindenstrauss lemma to analyze how the geometry of data changes after passing through randomly initialized convolutional neural networks, revealing preservation or contraction depending on the network type and input data.

Contribution

It introduces a geometric framework for CNNs, showing that linear CNNs preserve geometry while ReLU CNNs contract angles, with behavior depending on input data nature.

Findings

Linear CNNs preserve data geometry as per Johnson--Lindenstrauss lemma.

ReLU CNNs cause angle contraction, especially for Gaussian inputs.

Natural images' geometry is preserved after one CNN layer.

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.