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 of RD, where 0 < p ∈ R. Unlike other lpminimization problems, this minimization is nonconvex 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 lpminimization 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 lpminimization 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.
Bibliographical noteFunding Information:
This work was supported by NSF grants DMS-09-15064 and DMS-09-56072. It is inspired by our collaboration with Arthur Szlam on efficient and fast algorithms for hybrid linear modeling, which apply geometric minimization. We thank the anonymous reviewer for many insightful comments and suggestions that significantly improved the presentation of this work, John Wright for referring us to [, ] as well as for relevant questions which we address in Sect. and Vic Reiner, Stanislaw Szarek, and J. Tyler Whitehouse for commenting on an earlier version of this manuscript. Thanks to the Institute for Mathematics and its Applications (IMA) for holding a workshop on multi-manifold modeling that GL co-organized and TZ participated in.
© 2014, Springer Science+Business Media New York.
- Best approximating subspace
- Geometric probability
- Hybrid linear modeling
- Optimization on the Grassmannian
- Robust statistics