In the co-sparse analysis model a set of filters is applied to a signal out of the signal class of interest yielding sparse signal 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 efficiently learn an analysis operators with separable structures, which incorporates an efficient step size selection. Numerical experiments are provided that link the sample complexity to the convergence speed of the SGD algorithm.
Dong @cite_12 propose a method that alternates between a hard thresholding operation of the co-sparse representation and an operator update stage where all rows of the operator are simultaneously updated using a gradient method on the sphere. Their target function has the form @math , where @math is the sparse representation of the signal @math .
@cite_12
2025948329
We consider the dictionary learning problem for the analysis model based sparse representation. A novel algorithm is proposed by adapting the synthesis model based simultaneous codeword optimisation (SimCO) algorithm to the analysis model. This algorithm assumes that the analysis dictionary contains unit Ł 2 -norm atoms and trains the dictionary by the optimisation on manifolds. This framework allows one to update multiple dictionary atoms in each iteration, leading to a computationally efficient optimisation process. We demonstrate the competitive performance of the proposed algorithm using experiments on both synthetic and real data, as compared with three baseline algorithms, Analysis K-SVD, analysis operator learning (AOL) and learning overcomplete sparsifying transforms (LOST), respectively.