Learning Sets and Subspaces
Alessandro Rudi, Guillermo D. Cañas, Ernesto De Vito, Lorenzo Rosasco · 2014
Alessandro Rudi∗ DIBRIS, Università degli Studi di Genova and LCSL, Massachusetts Institute of Technology, and Istituto Italiano di Tecnologia Guillermo D. Canas∗ Massachusetts Institute of Technology Ernesto De Vito∗ DIMA, Università degli Studi di Genova Lorenzo Rosasco∗ DIBRIS, Università degli Studi di Genova and LCSL, Massachusetts Institute of Technology, and Istituto Italiano di Tecnologia 15.1 Unsupervised Statistical Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 15.2 Subspace Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 15.2.1 Problem Definition and Notation . . . . . . . . . . . . . . . . . . . . . . . 340 15.2.2 Subspace Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 15.2.3 Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 15.2.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 15.2.5 Kernel PCA and Embedding Methods . . . . . . . . . . . . . . . . . . 343 15.2.6 Comparison with Previous Results in the Literature . . . 344 15.3 Set Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 15.3.1 Set Learning via Subspace Learning . . . . . . . . . . . . . . . . . . . . 345 15.3.2 Consistency Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 15.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 15.5 Sketch of the Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 15.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 We consider here the classic problem of support estimation, or learning a set from random samples, and propose a natural but novel approach to address it. We do this by investigating its connection with a seemingly distinct problem, namely subspace learning.