Manifold Learning from Unorganized High-dimensional Data Points

研究者和他的同事们使用谱方法来研究无监督学习和数据可视化中的一个具有挑战性的问题：给定一组无组织的数据点，可能带有来自底层流形的噪声，计算一组数据点相对于底层流形的全局坐标。 该方法的指导原则是，全球结构可以从仔细分析当地的相互作用。 其核心思想是利用加权PCA方法来探索非线性流形上的切空间作为局部几何的表示，并通过计算邻域连接矩阵的部分特征分解来对这些局部切空间进行全局分解，从而获得流形的全局结构. 本文主要研究了以下几个方面：1）流形学习中有效的局部几何和拓扑结构; 2）局部采样密度、噪声水平、流形的正则性和局部曲率结构之间的相互作用及其对流形学习精度的影响; 3）局部光滑方法使流形学习对噪声和离群点具有更强的鲁棒性; 4）与无限混合模型、单类支持向量机和水平集方法的联系; 5）大规模流形学习问题的高效可扩展算法。 随着现代计算技术的发展，如何从海量数据中提取有用的信息是一个具有挑战性的问题。 在各种应用程序中产生的数据往往有成千上万的属性，导致了以“维数灾难”为特征的问题。“然而，在许多应用中，高维数据点是由几个固有的自由度控制的。想想一组描绘一张水平倾斜的脸的图像：这些图像是高维的，但图像集的固有自由度只是一个，代表了脸的旋转角度。 这个建议的重点是几何，统计和计算方法的发展，提取那些潜在的内在自由度，通过modelingthe数据点集作为样本从非线性流形，并发现全局结构的流形从仔细分析当地的相互作用。 研究结果、算法和技术在生物信息学中有着直接的应用，特别是用于检测和区分疾病和疾病类型的基因表达分析;用于基于外观的建模，用于使用监控摄像机生成的视频序列进行人员检测;帮助情报分析人员通过更有效的文本文档集建模来筛选大量的文本信息;以及在第二语言学习中加强计算机支持的协作学习和绩效评估。

The investigator and his colleagues use spectral methods toinvestigate a challenging problem in unsupervised learning anddata visualization: given a set of unorganized data pointssampled possibly with noise from an underlying manifold, computea set of global coordinates of the data points with respect tothe underlying manifold. The approach is guided by the generalprinciple that global structures can emerge from careful analysisof local interactions. The central idea is the exploration oftangent spaces as representations of local geometry on anonlinear manifold using weighted PCA, and the global alignmentof those local tangent spaces to obtain the global structure ofthe manifold by way of computing a partial eigendecomposition ofthe neighborhood connection matrix. The study focuses on thefollowing areas: 1) effective local geometric and topologicalstructures for manifold learning; 2) the interaction of the localsampling density, noise level, the regularity of the manifold andthe local curvature structure, and their effects on the accuracyof manifold learning; 3) local smoothing methods to make manifoldlearning more robust to noise and outliers; 4) connections withinfinite mixture models, one-class support vector machines andlevel set methods; 5) efficient and scalable algorithms forlarge-scale manifold learning problems. With the advancement in modern computing technology, we arefaced with the challenging problems of extracting usefulinformation from vast amounts of data. The data generated in avariety of applications tend to have tens of thousands ofattributes, leading to the problem characterized by "curse ofdimensionality." In many applications, however, high-dimensionaldata points are governed by a few intrinsic degrees of freedom.Think of the set of images depicting a horizontally rotatingface: the images are high-dimensional but the intrinsic degree offreedom of the image set is simply one, representing the rotationangle of the face. The focus of this proposal is the developmentof geometric, statistical, and computational methods forextracting those latent intrinsic degrees of freedom by modelingthe set of data points as samples from nonlinear manifolds, andthe discovery of the global structure of the manifolds fromcareful analysis of local interactions. The results, algorithms,and techniques developed have immediate applications inbioinformatics, especially for gene expression analysis fordetecting and distinguishing diseases and disease types; inappearance-based modeling for people detection using videosequences generated from surveillance cameras; in helpingintelligence analysts to sift through large amount of textualinformation by more efficient modeling of text documentcollections; and in enhancing computer-supported collaborativelearning and performance measurement in second language learning.