Adaptive signal representations: How much is too much?

Adaptive signal representations in overcomplete libraries of waveforms have been very popular. One naturally expects that in searching through a large number of signal representations for noisy data, one is at risk of identifying apparent structure in the data which turns out to be spurious, noise-induced artifacts. We show how to use penalties based on the logarithm of library complexity to temper the search, preventing such spurious structure, and giving near-ideal behavior.