基于子空间聚类的鲁棒高斯混合模型参数估计研究

人工智能和机器学习技术的迅速发展和广泛应用极大地促进了工业生产，改善了社会生活。以高斯混合模型为代表的概率模型作为机器学习的技术基础，具有重要研究意义。本项目计划采用新技术和新理论研究高斯混合模型的参数估计问题，设计鲁棒的有完备理论保障的新方法，并应用于高光谱图像分割等任务中，实现从理论基础向工程应用的转化。主要研究：构建无约束高斯混合模型与子空间簇模型的转换关系；在此基础上利用近年发展起来的子空间聚类技术，解决现有估计方法依赖初值和先验的固有难题；发展新的谱聚类分析理论，填补关于参数估计精度的完备理论分析的空白；基于随机投影等普适降维方法，探索面向高维数据的低复杂度方法及压缩维度比的信息极限；针对高光谱图像分割等应用，设计高效算法并在真实数据集上进行验证。本项目的创新性在于提出利用子空间聚类技术设计鲁棒高斯混合模型参数估计方法，发展谱聚类分析理论为参数估计精度提供理论保证。

In recent years, the rapid development of machine learning and artificial intelligence has significantly improved industrial productivity and facilitated our daily life. Among the foundational techniques in machine learning, probability models, for which the Gaussian Mixture Model (GMM) is the most widely studied representative, are found to be of great importance. In this project, we will address the problem of parameter estimation in GMM by utilizing new methods and new theories developed very recently. We aim at designing a new accurate and robust method for parameter estimation and providing theoretical analysis for the new method. Subsequently, the new method will be applied to hyperspectral image segmentation, fulfilling our goal of building a solid theoretical foundation and designing practical algorithms. The main contents of this project include: establishing the convertibility between unconstrained GMM and the Union of Subspace (UoS) model; based on their equivalence, designing a new robust parameter estimation method for GMM by utilizing the powerful sparse subspace clustering technique to solve the current problem of highly depending on initial values and prior knowledge; providing further analysis for the new method by developing the theory of spectral clustering for random geometric graph; improving the accuracy and computational efficiency of the proposed method by introducing weakly-convex optimization and random projection; finally, based on the proposed method, designing practical algorithms for applications such as hyperspectral image segmentation, and validating the algorithms on real-world datasets. The novelty of this project can be summarized as following: proposing a new robust approach to the problem of GMM parameter estimation via subspace clustering technique, enriching the theory of spectral clustering, and providing a solid theoretical guarantee for the new approach.