RANDOM MESH PROJECTORS FOR INVERSE PROBLEMS
We propose a new learning-based approach to solve ill-posed inverse problems in imaging. We address the case where ground truth training samples are rare and the problem is severely ill-posed-both because of the underlying physics and because we can only get few measurements. This setting is common in geophysical imaging and remote sensing. We show that in this case the common approach to directly learn the mapping from the measured data to the reconstruction becomes unstable. Instead, we propose to first learn an ensemble of simpler mappings from the data to projections of the unknown image into random piecewise-constant subspaces. We then combine the projections to form a final reconstruction by solving a deconvolution-like problem. We show experimentally that the proposed method is more robust to measurement noise and corruptions not seen during training than a directly learned inverse.
This paper describes a novel method for solving inverse problems in imaging.
The basic idea of this approach is use the following steps:
1. initialize with nonnegative least squares solution to inverse problem (x0)
2. compute m different projections of x0
3. estimate x from the m different projections by solving "reformuated" inverse problem using TV regularization.
The learning part of this algorithm is in step 2, where m different convolutional neural networks are used to learn m good projections. The projections correspond to computing a random Delaunay triangulation over the image domain and then computing pixel averages within each triangle. It's not clear exactly what the learning part is doing, i.e. what makes a "good" triangulation, why a CNN might accurately represent one, and what the shortcomings of truly random triangulations might be.
More specifically, for each projection the authors start with a random set of points in the image domain and compute a Delaunay triangulation. They average x0 in each of the Delaunay triangles. Then since the projection is constant on each triangle, the projection into the lower-dimensional space is given by the magnitude of the function over each of the triangular regions. Next they train a convolutional neural network to approximate the above projection. The do this m times. It's not clear why the neural network approximation is necessary or helpful. 
Empirically, this method outperforms a straightforward use of a convolutional U-Net to invert the problem.
The core novelty of this paper is the portion that uses a neural network to calculate a projection onto a random Delaunay triangulation. The idea of reconstructing images using random projections is not especially new, and much of the "inverse-ness" of the problem here is removed by first taking the pseudoinverse of the forward operator and applying it to the observations. Then the core idea at the heart of the paper is to speed up this reconstruction using a neural network by viewing the projection onto the mesh space as a set of special filter banks which can be learned.
At the heart of this paper is the idea that for an L-Lipschitz function f : R^k → R the sample complexity
is O(L^k), so the authors want to use the random projections to essentially reduce L. However, the Cooper sample complexity bound scales with k like k^{1+k/2}, so the focus on the Lipschitz constant seems misguided.
This isn't damning, but it seems like the piecewise-constant estimators are a sort of regularizer, and that's where we
really get the benefits.
The authors only compare to another U-Net, and it's not entirely clear how they even trained that U-Net. It'd be nice to see if you get any benefit here from their method relative to other approaches in the literature, or if this is just better than inversion using a U-Net. Even how well a pseudoinverse does would be nice to see or TV-regularized least squares.
Practically I'm quite concerned about their method requiring training 130 separate convolutional neural
nets. The fact that all the different datasets give equal quality triangulations seems a bit odd, too. Is
it possible that any network at all would be okay? Can we just reconstruct the image from regression
on 130 randomly-initialized convolutional networks? 
The proposed method isn't bad, and the idea is interesting. But I can't help but wonder whether it works just because what we're doing is denoising the least squares reconstruction, and regression on many random projections might be pretty good for that. Unfortunately, the experiments don't help with developing a deeper understanding.