2011/12/01 by Bing Li, Andreas Artemiou, Lexin Li · 1 citation
Computer Science · Engineering · Mathematics · #Algorithm #Applied mathematics #Artificial intelligence #Combinatorics #Computer science #Context (archaeology) #Control Systems and Identification #Dimension (graph theory) #Dimensionality reduction #Estimator #Face and Expression Recognition #Fault Detection and Control Systems #Hilbert space #Hyperplane #Kernel (algebra) #Kernel method #Mathematical optimization #Mathematics #Nonlinear system #Pure mathematics #Reduction (mathematics) #Reproducing kernel Hilbert space #Statistics #Support vector machine #math.ST #stat.TH
paper · pdf · doi:10.1214/11-aos932
published as Annals of Statistics 2011, Vol. 39, No. 6, 3182-3210 · Published in at http://dx.doi.org/10.1214/11-AOS932 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2011/12/01 · arxiv created 2012/03/13 · arxiv updated 2012/03/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, √n-consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hilbert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.