2011/04/30 by Gilad Lerman, Teng Zhang · 54 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Data point #Energy minimization #Linear subspace #Medical Image Segmentation Techniques #Minification #Outlier #Statistical Methods and Inference #math.ST #stat.ML #stat.TH
paper · pdf · doi:10.1214/11-aos914
published in The Annals of Statistics 39(5) (Institute of Mathematical Statistics) · Published in at http://dx.doi.org/10.1214/11-AOS914 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2011/10/01 · arxiv created 2012/02/01 · arxiv updated 2015/03/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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.