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.

arXiv comment11 pages double column, 4 figures, 3 tables

References in corpus (8)

• NESTA: A Fast and Accurate First-order Method for Sparse Recovery
• Stochastic gradient descent on Riemannian manifolds
• The Cosparse Analysis Model and Algorithms
• Analysis Operator Learning and Its Application to Image Reconstruction
• Insights into analysis operator learning: From patch-based sparse models to higher-order MRFs
• Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling
• Learning ${\ell }_1$-based analysis and synthesis sparsity priors using bi-level optimization
• Analysis Based Blind Compressive Sensing

Cited by in corpus (4)

• ${\ell }^1$-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?
• Image Fusion With Cosparse Analysis Operator
• Convolutional Analysis Operator Learning: Dependence on Training Data
• Online and Stable Learning of Analysis Operators