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October 2011 Robust recovery of multiple subspaces by geometric lp minimization
Gilad Lerman, Teng Zhang
Ann. Statist. 39(5): 2686-2715 (October 2011). DOI: 10.1214/11-AOS914


We assume i.i.d. data sampled from a mixture distribution with K components along fixed d-dimensional linear subspaces and an additional outlier component. For p > 0, we study the simultaneous recovery of the K fixed subspaces by minimizing the lp-averaged distances of the sampled data points from any K subspaces. Under some conditions, we show that if 0 < p ≤ 1, then all underlying subspaces can be precisely recovered by lp minimization with overwhelming probability. On the other hand, if K > 1 and p > 1, then the underlying subspaces cannot be recovered or even nearly recovered by lp minimization. The results of this paper partially explain the successes and failures of the basic approach of lp energy minimization for modeling data by multiple subspaces.


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Gilad Lerman. Teng Zhang. "Robust recovery of multiple subspaces by geometric lp minimization." Ann. Statist. 39 (5) 2686 - 2715, October 2011.


Published: October 2011
First available in Project Euclid: 22 December 2011

zbMATH: 1232.62097
MathSciNet: MR2906883
Digital Object Identifier: 10.1214/11-AOS914

Primary: 62G35 , 62H30 , 68Q32

Keywords: clustering , Detection , geometric probability , High-dimensional data , hybrid linear modeling , multiple subspaces , optimization on the Grassmannian , robustness

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 5 • October 2011
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