基于稀疏解析模型和低秩约束的分布式字典学习方法研究

Distributed dictionary learning can overcome the limitations of traditional dictionary learning which requires centralized training of data. However, existing distributed dictionary learning methods are all based on the sparse synthesis model with high computational complexity and do not fully exploit the essential features of the dictionary, which restricts the representation accuracy of the dictionary. Based on sparse analysis model and low-rank constraint, this proposal intends to study the mathematical modeling, optimization, performance analysis and application of distributed dictionary learning. Firstly, sparse analysis model is employed to formulate the overall sparse representation error and sparsity of distributed signals. The dictionary at each node is divided into a node-specific part and a global-shared part, and a low-rank constraint is introduced to the global-shared part to reflect its intrinsically low-dimensional characteristic. In addition, consensus constraints are introduced to realize information sharing among distributed nodes. Secondly, the non-convex optimization problem corresponding to the distributed analysis dictionary learning model under low-rank constraint is transformed into an unconstrained convex optimization problem, and based on convex optimization theory it is then decomposed into a series of sub-problems which are easier to be addressed. Finally, performance and theoretical analysis of the distributed dictionary learning method is carried out through simulation experiments, and this method will be applied to distributed sonar image denoising problem. This research can help improve the sparse representation performance of distributed dictionary learning and provide theoretical support and technical guidance for promoting its applications.

分布式字典学习可克服传统字典学习需要对数据进行集中式训练的局限性，而已有的分布式字典学习方法均基于复杂度较高的稀疏综合模型且未充分挖掘字典的本质特征，制约了字典的稀疏表示精度。本项目拟基于稀疏解析模型和低秩约束对分布式字典学习的数学建模、优化求解、性能分析和应用展开研究。首先，基于稀疏解析模型对分布式信号的总体稀疏表示误差和稀疏性进行建模，将单个节点字典划分为节点特定部分和全局共享部分，且对全局共享部分引入低秩约束以反映其内在低维性，并通过一致性约束实现节点信息共享；其次，将低秩约束下分布式解析字典学习模型对应的非凸优化问题变换为无约束凸优化问题，并结合凸优化理论将其分解为一系列易于求解的优化子问题；最后，通过仿真实验对分布式字典学习方法进行性能和理论分析，并将其应用于分布式的声呐图像去噪问题。项目研究有助于提高分布式字典学习的稀疏表示性能，为促进其应用提供理论支持和技术指导。

分布式字典学习可直接利用分布式存储的训练数据，能够避免大规模或保密性数据的直接交换，而已有分布式字典学习方法基于稀疏综合模型，且未充分挖掘字典特征，限制了所学字典的表示性能。针对这些问题，本项目对基于稀疏解析模型和低秩约束的分布式字典学习方法展开研究。首先，对低秩约束的数学建模进行研究，将基于非凸函数的低秩模型与稀疏先验相结合，提出一种基于低秩-稀疏混合模型的分解算法；其次，引入基于局部字典的信息一致性约束，以实现节点信息共享，并基于此构建分布式解析字典学习模型和相应的学习算法；最后，将已有集中式解析字典学习的数学模型扩展到分布式环境下，并采用扩散策略实现全局解析字典的分布式学习，并将算法应用于分布式图像去噪问题。项目对所提算法进行了充分的验证与测试，实验结果显示项目所提出的分布式解析字典学习算法可有效利用分布式数据，学习所得到的字典性能远远优于局部字典学习且能逼近于集中式字典学习结果。围绕项目相关研究内容，已发表SCI论文6篇，EI会议论文4篇，申请并公开发明专利3项，授权发明专利1项。项目研究成果丰富了分布式字典学习的数学模型和算法实现，推动了其在分布式图像去噪和图像恢复等领域的应用。