In the co-sparse analysis model a set of filters is applied to a signal out
of the signal class of interest yielding sparse filter responses. As such, it
may serve as a prior in inverse problems, or for structural analysis of signals
that are known to belong to the signal class. The more the model is adapted to
the class, the more reliable it is for these purposes. The task of learning
such operators for a given class is therefore a crucial problem. In many
applications, it is also required that the filter responses are obtained in a
timely manner, which can be achieved by filters with a separable structure. Not
only can operators of this sort be efficiently used for computing the filter
responses, but they also have the advantage that less training samples are
required to obtain a reliable estimate of the operator. The first contribution
of this work is to give theoretical evidence for this claim by providing an
upper bound for the sample complexity of the learning process. The second is a
stochastic gradient descent (SGD) method designed to learn an analysis operator
with separable structures, which includes a novel and efficient step size
selection rule. Numerical experiments are provided that link the sample
complexity to the convergence speed of the SGD algorithm.Comment: 11 pages double column, 4 figures, 3 table
