Manifold Learning from Unorganized High-dimensional Data Points

这位研究人员和他的同事们使用光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：给定一组可能带有来自底层流形的噪声采样的无组织数据点，计算数据点相对于底层流形的全局坐标集。这种方法的指导原则是，全球结构可以从对局部相互作用的仔细分析中产生。其核心思想是利用加权主成分分析来探索作为非线性流形上局部几何表示的切线空间，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线空间进行全局对齐以获得流形的全局结构。1)有效的局部几何拓扑结构用于流形学习；2)局部采样密度、噪声水平、流形的正则性和局部曲率结构的相互作用及其对流形学习精度的影响；3)局部平滑方法使流形学习对噪声和离群点更健壮；4)与无限混合模型、一类支持向量机和水平集方法的连接；5)大规模流形学习问题的高效和可扩展的算法。随着现代计算技术的进步，我们面临着从海量数据中提取有用信息的挑战性问题。在各种应用程序中生成的数据往往具有数万个属性，导致了以“维度诅咒”为特征的问题。然而，在许多应用中，高维数据点受几个固有自由度的支配。考虑到描绘水平旋转人脸的图像集：图像是高维的，但图像集的固有自由度仅为1，表示人脸的旋转角度。这一建议的重点是发展几何、统计和计算方法，通过将数据点集合建模为来自非线性流形的样本来提取潜在的内在自由度，并通过仔细分析局部相互作用来发现流形的全局结构。开发的结果、算法和技术立即应用于生物信息学，特别是用于检测和区分疾病和疾病类型的基因表达分析；使用监控摄像机生成的视频序列进行基于外表的人检测建模；帮助情报分析员通过更有效的文本文档集合建模来筛选大量文本信息；以及在第二语言学习中增强计算机支持的协作学习和表现测量。

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.