lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers

by   Gilad Lerman, et al.

We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the lp-averaged distances of data points from d-dimensional subspaces, where p>0. Unlike other lp minimization problems, this minimization is non-convex for all p>0 and thus requires different methods for its analysis. We show that if 0<p<=1, then for any fraction of outliers the most significant subspace can be recovered by lp minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces the most significant subspace can be nearly recovered by lp minimization for any 0<p<=1 with an error proportional to the noise level. On the other hand, if p>1 and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.


page 1

page 2

page 3

page 4


Probabilistic Recovery of Multiple Subspaces in Point Clouds by Geometric lp Minimization

We assume data independently sampled from a mixture distribution on the ...

Robust recovery of multiple subspaces by geometric l_p minimization

We assume i.i.d. data sampled from a mixture distribution with K compone...

Exact Subspace Segmentation and Outlier Detection by Low-Rank Representation

In this work, we address the following matrix recovery problem: suppose ...

Subspace approximation with outliers

The subspace approximation problem with outliers, for given n points in ...

List Decodable Subspace Recovery

Learning from data in the presence of outliers is a fundamental problem ...

Dual Principal Component Pursuit

We consider the problem of outlier rejection in single subspace learning...

A Novel M-Estimator for Robust PCA

We study the basic problem of robust subspace recovery. That is, we assu...

Please sign up or login with your details

Forgot password? Click here to reset