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

TL;DR

This paper investigates the problem of recovering the most significant subspace from data with multiple subspaces and outliers by using lp minimization, showing conditions under which recovery is possible or impossible.

Contribution

It introduces a novel analysis of lp minimization for subspace recovery, demonstrating its effectiveness for 0<p<=1 and limitations for p>1.

Findings

For 0<p<=1, the most significant subspace can be recovered with high probability despite outliers.

Adding small noise allows near recovery of the subspace proportional to noise level.

For p>1, the most significant subspace cannot be reliably recovered when multiple subspaces are present.

Abstract

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.