Compression is cognition. Gerry Wolf has also been saying this for many years. I haven't studied his work closely enough or recently enough to comment on it.
Lots of people promote the compression idea. But as far as I know, with the exception of the Hutter Prize, no one has actually attempted to do large scale compression, in any kind of systematic way. People in SNLP talk about language modeling, but they are not systematic (no shared benchmarks, no competitions, etc). No one in computer vision attempts to do large scale image compression.
There must be quite a few undergrad/graduate/post-doc/???-level researchers on LessWrong. I'm interested in hearing about your work. I'll post about myself in the comments.