The Sample Complexity of Dictionary Learning

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 $O\left(\sqrt{np\ln(m λ)/m}\right)$, 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 $O(\sqrt{np\ln(m k)/m})$ under an assumption on the closeness to orthogonality of the dictionary (low Babel function). We further show that this assumption holds for 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.