We consider the problem of learning a low-dimensional signal model from a collection of training samples. The mainstream approach would be to learn an overcomplete dictionary to provide good approximations of the training samples using sparse synthesis coefficients. This famous sparse model has a less well known counterpart, in analysis form, called the cosparse analysis model. In this new model, signals are characterised by their parsimony in a transformed domain using an overcomplete (linear) analysis operator. We propose to learn an analysis operator from a training corpus using a constrained optimisation framework based on L1 optimisation. The reason for introducing a constraint in the optimisation framework is to exclude trivial solutions. Although there is no final answer here for which constraint is the most relevant constraint, we investigate some conventional constraints in the model adaptation field and use the uniformly normalised tight frame (UNTF) for this purpose. We then derive a practical learning algorithm, based on projected subgradients and Douglas-Rachford splitting technique, and demonstrate its ability to robustly recover a ground truth analysis operator, when provided with a clean training set, of sufficient size. We also find an analysis operator for images, using some noisy cosparse signals, which is indeed a more realistic experiment. As the derived optimisation problem is not a convex program, we often find a local minimum using such variational methods. Some local optimality conditions are derived for two different settings, providing preliminary theoretical support for the well-posedness of the learning problem under appropriate conditions.

本研究提出了基于L1优化的受限极小化学习方法，用于从训练语料库中学习一个分析算子，通过投影次梯度和Douglas-Rachford分割技术，对足够大小的干净训练集进行稳健回收地真实分析算子，同时在一些噪声cosparse信号的帮助下，为图像找到一个分析算子。

基于约束过完备分析算子学习的余稀信号建模