Manifold Learning from Unorganized High-dimensional Data Points

研究者和他的同事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：给定一组无组织的数据点，可能从底层流形中采样并带有噪声，计算一组数据点相对于底层流形的全局坐标。该方法遵循的一般原则是，全球结构可以从对局部相互作用的仔细分析中产生。中心思想是利用加权PCA探索作为非线性流形局部几何表示的切空间，并通过计算邻域连接矩阵的部分特征分解，对这些局部切空间进行全局对准，以获得流形的整体结构。本研究主要集中在以下几个方面：1)流形学习的有效局部几何和拓扑结构；2)局部采样密度、噪声水平、流形的规律性与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异常值具有更强的鲁棒性；4)无限混合模型的连接、一类支持向量机和水平集方法；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进步，我们面临着从海量数据中提取有用信息的难题。在各种应用程序中生成的数据往往具有成千上万的属性，从而导致以“维度诅咒”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自由度控制的。考虑一组描绘一张水平旋转的脸的图像：这些图像是高维的，但图像集的固有自由度只是1，表示脸的旋转角度。本提案的重点是发展几何、统计和计算方法，通过对非线性流形的数据点集建模作为样本来提取这些潜在的内在自由度，并通过仔细分析局部相互作用来发现流形的全局结构。所开发的结果、算法和技术在生物信息学中具有直接的应用，特别是用于检测和区分疾病和疾病类型的基因表达分析；利用监控摄像机生成的视频序列进行基于隐身的人物检测建模通过对文本文档集合进行更有效的建模，帮助情报分析人员筛选大量文本信息；并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.