非凸非光滑低秩恢复模型与优化算法研究

With the theoretical research and extensive application of compressed sensing, low rank recovery has attracted much attention from researchers in the field of information science and optimization. Compared with the convex relaxation model, the nonconvex and nonsmooth model can better capture the essential properties of practical problems. Based on visual information representation and denoising as the basic application background, this project studies the key problems of exploring efficiently low rank recovery by using the low rank structure of information. In the light of prior knowledge, the classical low rank visual information recovery model is revisited under nonconvex and nonsmooth framework. The loss function and the low rank regularization term are generalized to the unified mathematical framework. The goal is to solve the difficulties cased from the nonlinearity of the low rank visual information, and to establish nonconvex and nonsmooth recovery models for visual information; In terms of the traditional optimization learning algorithm, the efficient algorithms are designed to decompose the original nonconvex and nonsmooth low rank recovery model into a few nonsmooth convex problems. In theory, we prove that the algorithm can get the optimal solution and study the validity and stability of the solution; The evaluation mechanism of the nonconvex and nonsmooth low rank recovery model is established, which includes the convergence of the model, the sensitivity of parameters, the complexity of time and space, and the verification in practical application. Efficient nonconvex and nonsmooth low rank recovery model not only helps to mine the semantic structure of visual information, but also helps to speed up the learning process.

低秩恢复因压缩感知理论研究及其广泛的应用，引起了信息科学与优化等领域研究者的极大关注。与凸松弛模型相比,非凸非光滑模型能够更好地刻画实际问题的本质属性。本项目以视觉信息表达与去噪为基本应用背景，利用信息间的低秩结构，对低秩恢复技术中的关键问题展开研究。根据先验知识，在非凸非光滑框架下重新审视经典的低秩视觉信息恢复模型，将损失函数与低秩正则项泛化到统一数学框架下，解决低秩视觉信息非线性所带来的难题，构建面向视觉信息的非凸非光滑恢复模型；借鉴传统优化学习算法，将原始非凸非光滑低秩恢复模型分解为多个非光滑的凸问题，在理论方面证明算法能够得到有效地最优解，并对解有效性与稳定性等问题进行研究；建立非凸非光滑低秩恢复模型的评价机制，包括模型收敛性、参数敏感性、时空间复杂度及在实际应用领域中的验证。高效的非凸非光滑低秩恢复模型与优化算法不仅有助于挖掘视觉信息潜在的语义结构，而且有利于加快学习进程。

伴随着移动数字电子设备广泛应用，各领域都在源源不断地产生图像与视频等视觉信息，项目组成员针对所获取带视觉信息构建三类模型与算法，主要包括面向多种验信息融合视觉信息恢复模型与优算法，面向稀疏与低秩结构信息约束的视觉表示模型与优化算法，面向非负与流形几何结构统一框架视觉表达模型与算法。基于深度学习和多模态数据，增加了多流形结构数据的深度学习方法和多模态数据的本质结构分析等方面的研究。取得了很好的效果，并应用到了图像恢复，人脸聚类，图像聚类，基于图的半监督学习，点云法向估计，基因表达数据等问题中。在本项目周期内，项目组成员共发表学术论文12篇，其中SCI检索8篇，发表国内核心期刊论文3篇，会议文章1篇，参加国内和国际会议29人次，培养硕士研究生毕业8人。作为参与人申请一项国家自然基金项目1项，项目负责人入选辽宁省高等院校创新人才支持计划项目。