The Sample Complexity of Dictionary Learning

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Daniel Vainsencher, Shie Mannor, Alfred M. Bruckstein ;
Proceedings of the 24th Annual Conference on Learning Theory, PMLR 19:773-788, 2011.

Abstract

A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from a given set of signals to be represented. Can we expect that the error in representing by such a dictionary a previously unseen signal from the same source will be of similar magnitude as those for the given examples?We assume signals are generated from a fixed distribution, and study these questions from a statistical learning theory perspective. We develop generalization bounds on the quality of the learned dictionary for two types of constraints on the coefficient selection, as measured by the expected L_2 error in representation when the dictionary is used.For the case of l_1 regularized coefficient selection we provide a generalization bound of the order of O\left(\sqrtnp\ln(m λ)/m\right), where n is the dimension, p is the number of elements in the dictionary, λis a bound on the l_1 norm of the coefficient vector and m is the number of samples, which complements existing results.For the case of representing a new signal as a combination of at most k dictionary elements, we provide a bound ofthe order O(\sqrtnp\ln(m k)/m) under an assumption on the closeness to orthogonality of the dictionary (low Babel function).We further show that this assumption holds for \em most dictionaries in high dimensions in a strong probabilistic sense.Our results also include bounds that converge as 1/m, not previously known for this problem.We provide similar results in a general setting using kernels with weak smoothness requirements.

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