A Note on the Regularity of Images Generated by Convolutional Neural Networks
This provides analytical evidence that basic L2 regularization in neural networks might cause over-smoothed outputs, which is an incremental insight for researchers in image processing and machine learning.
The paper analyzes the regularity of images generated by convolutional neural networks, showing that in an infinite-dimensional setting, these images are always continuous and sometimes continuously differentiable, which contradicts the common modeling of sharp edges as jump discontinuities.
The regularity of images generated by convolutional neural networks, such as the U-net, generative networks, or the deep image prior, is analyzed. In a resolution-independent, infinite dimensional setting, it is shown that such images, represented as functions, are always continuous and, in some circumstances, even continuously differentiable, contradicting the widely accepted modeling of sharp edges in images via jump discontinuities. While such statements require an infinite dimensional setting, the connection to (discretized) neural networks used in practice is made by considering the limit as the resolution approaches infinity. As practical consequence, the results of this paper in particular provide analytical evidence that basic L2 regularization of network weights might lead to over-smoothed outputs.