Learning Co-Sparse Analysis Operators with Separable Structures

Abstract--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. HE ability to sparsely represent signals has become standard practice in signal processing over the last decade. The commonly used synthesis approach has been extensively investigated and has proven its validity in many applications. Its closely related counterpart, the co-sparse analysis approach, was at first not treated with as much interest. In recent years this has changed and more and more work regarding the application and the theoretical validity of the co-sparse analysis model has been published. Both models assume that the signalss of a certain class are (approximately) contained in a union of subspaces. In the synthesis model, this reads as s Dx, x is sparse. Personal use of this material is permitted.