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Svmtorch: Support vector machines for large-scale regression problems. Bayesian support vector regression using a unified loss function. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(5):572-575, 1998. Intrinsic dimensionality estimation with optimally topology preserving maps. Hinging hyperplanes for regression, classification, and function approximation. In Advances in Neural Information Processing Systems 15, pages 961-968. In European Conference on Machine Learning, pages 47-59, 2005. Nonrigid embeddings for dimensionality reduction. Laplacian eigenmaps for dimensionality reduction and data representation. Universal approximation bounds for superpositions of a sigmoidal function. An optimal algorithm for approximate nearest neighbor searching. Finally, we show competitive function approximation results on real data. In addition, we compare our approach with several leading methods for manifold learning at the task of measuring geodesic distances. Quantitative results on the estimation of local manifold structure using ground truth data are presented. Moreover, these local dimensionality and structure estimates enable us to measure geodesic distances and perform nonlinear interpolation for data sets with varying density, outliers, perturbation and intersections, that cannot be handled by state-of-the-art methods. Analyzing the estimated local structure at the inputs, we are able to obtain reliable dimensionality estimates at each instance, instead of a global estimate for the entire data set. Tensor voting is a perceptual organization framework that has mostly been applied to computer vision problems. For this purpose we employ a novel formulation of tensor voting, which allows an N-D implementation. Unlike conventional manifold learning, we do not perform dimensionality reduction, but instead perform all operations in the original input space. Under our approach, manifolds in high-dimensional spaces are inferred by estimating geometric relationships among the input instances. We address instance-based learning from a perceptual organization standpoint and present methods for dimensionality estimation, manifold learning and function approximation.
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