Noisy Sparse Subspace Clustering with dimension reduction YI
Noisy Sparse Subspace Clustering with dimension reduction YI NING WA NG , YU-XIANG WANG, AARTI SINGH MACHINE LEARNING DEPARTMENT CARNEGIE MELLON UNIVERSITY 1
Subspace Clustering 2
Subspace Clustering Applications Motion Trajectories tracking 1 1 (Elhamifar and Vidal, 2013), (Tomasi and Kanade, 1992) 3
Subspace Clustering Applications Face Clustering 1 Network hop counts, movie ratings, social graphs, … 1 (Elhamifar and Vidal, 2013), (Basri and Jacobs, 2003) 4
Sparse Subspace Clustering e s s l n Fa tio o N nec n o C 5
Sparse Subspace Clustering Noiseless data Noisy data 6
SSC with dimension reduction 7
SSC with dimension reduction 8
Main Result 9
Proof sketch ◦ Review of deterministic success conditions for SSC (Soltanolkotabi and Candes, 12)(Elhamifar and Vidal, 13) ◦ Subspace incoherence ◦ Inradius ◦ Analyze perturbation under dimension reduction ◦ Main results for noiseless and noisy cases. 10
Review of SSC success condition Lasso SSC formulation Dual problem of Lasso SSC 11
Review of SSC success condition ◦ Inradius ◦ Characterzing inner-subspace data point distribution 12
Review of SSC success condition 13
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Main idea: linear operator transforms a ball to an ellipsoid 15
Main result Regularization Error of approximate parameter isometry of Lasso SSC required even for noiseless problem 16
Simulation results (Hopkins 155) 17
Conclusion Questions? 18
References ◦ M. Soltanolkotabi and E. Candes. A Geometric Analysis of Subspace Clustering with Outliers. Annals of Statistics, 2012. ◦ E. Elhamifar and R. Vidal. Sparse Subspace Clustering: Algorithm, Theory and Applications. IEEE TPAMI, 2013 ◦ C. Tomasi and T. Kanade. Shape and Motion from Image Streams under Orthography. IJCV, 1992. ◦ R. Basri and D. Jacobs. Lambertian Reflection and Linear Subspaces. IEEE TPAMI, 2003. ◦ Y. -X. , Wang and H. , Xu. Noisy Sparse Subspace Clustering. ICML, 2013. 19
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