A Johnson--Lindenstrauss Framework for Randomly Initialized CNNs
This work provides theoretical insights into the initialization behavior of CNNs, which is incremental as it extends known results from fully-connected networks to convolutional architectures.
The authors tackled the problem of understanding how the geometric representation of data changes through randomly initialized layers in convolutional neural networks (CNNs), showing that linear CNNs preserve angles as per the Johnson–Lindenstrauss lemma, while ReLU CNNs contract angles depending on input type, with natural images preserving geometry and Gaussian inputs contracting similarly to fully-connected networks.
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.