Manifold Learning from Unorganized High-dimensional Data Points

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Zha 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.