The crazy title up there means “recovering a 3D model of a molecule, using multiple 2D images/projections”. This was a talk at MIT by Yoel Shkolnisky. The talk was really neat, but the bottom line is this. From a CS point of view he was solving the following question. We have n points that (we know) come from some finite metric. We are given the pairwise distances (+ noise) between points that are within each other’s vicinity (in that metric). The goal is to reconstruct the metric. I’ve phrased this problem a little more generally then the form in which it is addressed in Yoel’s work.

The cool thing is that Yoel et al. solve this problem using a very natural appeal to graph Laplacians. This is found in a technical report on his page. Amit Singer (a collaborator) additionally has a paper on using these techniques to recover the global metric of sensor networks, based on local distance information. This is found in the form of a paper on Amit’s webpage.

There might be an opportunity for a new paper lurking in there. In particular, it is interesting to ask under what noise conditions (and noise can be correlated in real-world examples) is the recovery good. Having heard the talk, I believe that the answer to this question will also depend on the type of metric being recovered, i.e. it may depend on its doubling dimension e.g.

Oh, I should also mention that Yoel seems to have some cool papers involving the discrete Radon transform. This is probably a good place to read about it, while avoiding the continuous lingo.