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Dentin comments on A Fervent Defense of Frequentist Statistics - Less Wrong
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Compressed sensing isn't even magic, if you're halfway versed in signal processing. I understood compressed sensing within 30 seconds of hearing a general overview of it, and there are many related analogs in many fields.
The convex optimization guys I know are all rather impressed by compressed sensing- but that may be because they specialize in doing L1 and L2 problems, and so compressed sensing makes the things they're good at even more important.