Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces
The Journal of Machine Learning Research
Semi-Supervised Learning on Riemannian Manifolds
Machine Learning
Learning Coordinate Covariances via Gradients
The Journal of Machine Learning Research
Estimation of Gradients and Coordinate Covariation in Classification
The Journal of Machine Learning Research
Dimensionality Reduction of Multimodal Labeled Data by Local Fisher Discriminant Analysis
The Journal of Machine Learning Research
Regularization on Graphs with Function-adapted Diffusion Processes
The Journal of Machine Learning Research
Semi-Supervised Learning
Towards a theoretical foundation for laplacian-based manifold methods
COLT'05 Proceedings of the 18th annual conference on Learning Theory
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The problems of dimension reduction and inference of statistical dependence are addressed by the modeling framework of learning gradients. The models we propose hold for Euclidean spaces as well as the manifold setting. The central quantity in this approach is an estimate of the gradient of the regression or classification function. Two quadratic forms are constructed from gradient estimates: the gradient outer product and gradient based diffusion maps. The first quantity can be used for supervised dimension reduction on manifolds as well as inference of a graphical model encoding dependencies that are predictive of a response variable. The second quantity can be used for nonlinear projections that incorporate both the geometric structure of the manifold as well as variation of the response variable on the manifold. We relate the gradient outer product to standard statistical quantities such as covariances and provide a simple and precise comparison of a variety of supervised dimensionality reduction methods. We provide rates of convergence for both inference of informative directions as well as inference of a graphical model of variable dependencies.