The abstract is hard to parse for a layman. Anyone care to venture a translation for the non-ML wonks?

So I'm not an expert in the mathematics they're using specifically, but my high level understanding is that convolutional neural nets are able to extract features because they satisfy certain invariance properties. For example, a face is still a face even if it's shifted to the right, made a little smaller, or tilted a bit. Therefore, the feature of faces should be invariant to these transformations. This paper develops the mathematical framework to better help understand when such invariance properties pop up in a wider class of deep neural net architectures, and thereby when these architectures can identify broader classes of features.

Deep convolutional networks have enjoyed great success in image recognition and understanding in the last 3+ years (see the ImageNet competition winners). One of the main reasons people believe they are so powerful is due to translation, rotation, and scale invariance of the learned networks: the top layer coefficients are very similar (in terms of squared norm) regardless of the target object's location, size, and rotation in the image.

This paper basically shows that the structure of deep convolutional networks (specifically: shared convolutional kernels at each layer, subsampling / pooling across layers) are the main reason behind this behavior. Furthermore, the relative invariance increases with the depth (number of stacked convolutional layers).

They show this by first showing that a wide variety of convolutional kernels and pooling behaviors provide said invariance (the tools they use are from the world of graduate level calculus + analysis: wavelets and frames); they also show stability with regards to nonlinear deformation (even if the original image is swirled around / deformed a bit before shifting, rotating, scaling; the coefficients still don't change much). That's one major result of this paper.

Then they show that the invariance property is not tied to the kernels per se: you don't need the intermediate convolutions to be a wavelet, fourier transform, or whatever. This is important because neural networks don't use a fixed convolutional kernel - they learn one on the fly from training data. The theorem in this paper that shows the invariance is somewhat independent of the specific coefficients of the kernel and therefore general convolutional neural networks share these same properties.

That said, the learned kernels of the lowest layers of convolutional neural networks do tend to look like wavelets / curvelets / shearlets / etc, while the higher convolutional layers tend to look like higher level features. So neural networks' kernels are already close enough to the theoretical family of possible convolutional functions studied in this paper for the theorems to more or less apply.

Since it's impossible to prove that a Conv NN trained empirically on a dataset of natural images actually exactly learns wavelets or anything possible to study theoretically, it's nice to have a paper that says it's not the kernels themselves that are super important, but that the structure of pretty much any deep convolutional NN, is the important part.

Finally, the paper extends a neat way to think about convolutional neural networks (by extendint Mallat's initial work): that the layers represent functions which are invariant to group operations like rotation and translation, and stable with respect to deformation. This means you can study these types of neural networks using group theory (abstract algebra); an area of math where it's easy to gather insight about how things work and develop new algorithms that you can translate back to the real world.

CNNs are not rotation invariant.

Read the introduction of the actual paper.