Structured sparse learning has become a popular and mature research field. Among all structured sparse models, we found an interesting fact that most structured sparse properties could be captured by convolution operators, most famous ones being total variation and wavelet sparsity. This finding has naturally brought us to a generalization termed as Convolutional Sparsity. While this generalization bridges the convolution and sparse learning theory, we are able to propose a general, efficient, hyperparameter-free optimization algorithm framework for convolutional sparse models, thanks to the analysis theory of convolution operators. The convergence of the general, hyperparameter-free algorithm has been comprehensively analyzed, with a non-ergodic rate of O(1/ϵ2) and ergodic rate of O(1/ϵ), where ϵ is the desired accuracy. Extensive experiments confirm the superior performance of our general algorithm in various convolutional sparse models, even better than some application-specialistic algorithms.

In this paper, we propose an algorithm for missing value recovery of visual data such as image or video. These missing values may result from the corruption in acquisition process, or user-specified unexpected outliers. This problem exists in wide range of applications. We use the nuclear norm (NN) regularization to enforce the global consistency of the image, while the total variation (TV) regularization is used to encourage the locally consistent in image intensity domain. This model can be applied in very challenging scenarios, where only very small amount of data is available. However, it is very difficult to efficiently solve these two regularizations simultaneously by convex programming due to its composite structure and non-smoothness. To this end, we propose an efficient proximal-splitting algorithm for joint NN/TV minimization. The proposed algorithm is theoretically guaranteed to achieve a convergence rate of O(1/N) for N iterations, which is much faster than O(1/√N) by the black-box first-order method for solving the non-smooth optimization problem. In our experiments, we demonstrate the superior performance of our algorithm on image completion compared with seven state-of-the-art algorithms.