Today was my first day at the Simons Center for Data Analysis. I spent a good chunk of the morning discussing projects with Leslie Greengard (SCDA), the director of the place. We discussed cryogenic EM imaging of molecules, in which you get many shadows of the same molecule at different positions and orientations. The inference of the three-dimensional structure involves inferring also all the projection angles and offsets—or does it?. We discussed the problem of neuron spike sorting, where potential events observed from an array of electrodes (inserted in a brain) are assigned to neurons. How do you know if you got it right? Many algorithms are stable, but which are correct?
Later in the day, Mike O'Neil joined us and we discussed (among other things) non-negativity as a constraint on problem solving. This is an amazingly informative constraint. For example, if you are doing image reconstruction and you have an array of a thousand by a thousand pixels, the non-negativity constraint removes all but one part in two to the millionth power of your parameter space! That is more informative than any data set you could possibly get. Greengard didn't like this argument: It is a counting argument! And I was reminded of Phillip Stark's (Berkeley) objection to it: He said something to the effect of “No, the constraint is much stronger than that, because it applies everywhere in space, even where you haven't sampled, so it is really an infinite number of constraints”. Greengard showed us some magic results in diffractive imaging where it appears that the non-negativity constraint is doing a huge part of the heavy lifting. This all relates to things I have discussed previously with Bernhard Schölkopf and his team.