Big Numbers

PCMI (1) July 1, 2010

First, let me get it out of the way: Utah is beautiful. I’ve wanted to go to Utah ever since I read A Primer on Mapping Class Groups and realized that there are people doing cool math out west. It’s amazing up here. Liquid blue sky, sagebrush on the mountains, columbines and lupines and pine martens…
It’s actually so pretty that it tempts me to get too ambitious with the trail running.

It’s pretty busy, as I’m taking both undergraduate and graduate courses.

Highlights:

Jared Tanner’s course on compressed sensing is, as far as anyone knows, the first set of lectures on this for an undergraduate audience. We showed that sparse recovery is possible for matrices in general position (checking this property is impractical, as it takes exponential time, but it happens that random matrices are in general position with high probability.) We went on to deal with Haar and Fourier bases, and we’re now studying basis pursuit and coherence.

Richard Baraniuk gave some graduate lectures on compressed sensing, from more of an application perspective. He gave a visual illustration that I really liked of why l^1 minimization is a better way to find sparsity than l^2 minimization — the l^1 ball is an octahedron, and it’s likely to intersect a line of random angle close to a coordinate axis. The l^2 ball is a sphere, which is less “concentrated” around the axes, and so is likely to intersect a line of random angle farther from a coordinate axis — that is, less sparse.

I’m now starting a course with Anna Gilbert about sparse approximation — she’s going to take more of the computational complexity approach.

The other nice thing about PCMI is meeting people you’ve only heard of through their research. I went hiking (and then pizza-eating) with Arthur Szlam, whom I knew about because he got his PhD from Yale and does the same kind of computational harmonic analysis that I’d (ideally) like to do. Talking to him was great — he runs at the prodigious rate of about three big mathematical insights per beer.