  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 set of signals to be represented. Can we expect that the
representation found by such a dictionary for a previously unseen example from
the same source will have L_2 error of the same magnitude as those for the
given examples? We assume signals are generated from a fixed distribution, and
study this 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(sqrt(np log(m lambda)/m)), where n is the dimension, p is the
number of elements in the dictionary, lambda is 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 of the order O(sqrt(np log(m k)/m))
under an assumption on the level of 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 further yield fast
rates of order 1/m as opposed to 1/sqrt(m) using localized Rademacher
complexity. We provide similar results in a general setting using kernels with
weak smoothness requirements.
