机器学习中的低秩与稀疏矩阵逼近理论及算法研究

As an important development of sparse representation and compressed sensing, low rank matrix recovery has been viewed as a new effective method of data acquisition and reconstruction. High-dimensional data is often faced with some lost or corrupted with noises or obtained by a linear measurement. How to make use of the structure of the data to recover target matrix, has widely been used in signal processing, pattern recognition, artificial intelligence, and other fields. The research project focuses on the low rank and sparse matrix recovery. To summarize the existing algorithms, we present our low rank and sparse decomposition model based on graph Laplacian, and corresponding algorithm. Combining with the sparse regularized algorithm (e.g. Lasso、Elastic-Net algorithm) in learning theory, we propose the low rank and sparse matrix elastic-net regularization model. For matrix data learning problem, we research the convergence, consistency and learning rates of the matrix recovery algorithm, and analysis the theoretical foundation of these algorithms. We also discuss the inherent relations between matrix recovery and the best rank r approximation. This project belongs to the theory and application of reconstruction algorithms in low rank matrix recovery problem, belongs to the frontier research topic in cross discipline for the applied mathematics and engineering applications. The study is of important theoretical significance and practical value.

作为稀疏表示和压缩感知理论的重要发展，矩阵恢复问题已成为一种新型数据获取与重构的有效方法，高维数据矩阵面临着部分丢失或被污染情形下或通过某种线性运算得到，如何利用数据的结构恢复目标矩阵，逐渐在信号处理、模式识别、人工智能等领域有着广泛的应用。本项目的主要目标是，充分研究和总结目前已有的低秩矩阵恢复算法，构建基于图约束项的低秩与稀疏矩阵逼近重构算法；结合学习理论中的稀疏正则化算法（例如Lasso、Elastic-Net等），提出基于Elastic-Net正则化的低秩与稀疏矩阵逼近算法；针对矩阵型数据学习问题，研究算法的收敛性、相容性以及收敛速度估计，并分析其算法的理论基础；同时探讨矩阵恢复和最佳秩r逼近的内在联系。本项目属于低秩矩阵恢复中重构算法的理论及应用研究，是应用数学和工程应用领域交叉学科中的前沿研究课题,具有一定的理论价值和广泛的应用前景。

本课题研究了稀疏逼近和多核机器学习的相关理论和应用问题。具体内容有，在理论方面，我们提出了基于正则和空间模型的多尺度最小二乘支持向量机算法，得到了算法闭合解；提出了基于正则和空间模型的半监督分布式最小二乘支持向量机算法，降低了算法复杂度；利用覆盖数估计来建立多尺度算法学习率的方法推广到相关取样过程并解决算法有限维优化问题，得到了快速学习率；研究了一组欠定采样的稀疏信号恢复问题的RIP条件，放宽了分析字典学习中基于非凸优化算法的适用范围；提出了基于全变差正则化的自适应SAR图像超分辨率重构算法，设计了分裂Bregman算法框架。在实际应用方面，在光谱数据地物分类、SAR图像处理方面及电离层TEC值去噪方面，得到了一些有意义的研究成果。为函数逼近论和学习理论在实际应用中提供新的研究手段。