Robust recovery of multiple subspaces by geometric l_p minimization

TL;DR

This paper analyzes the effectiveness of l_p minimization for recovering multiple linear subspaces from data, showing it works well for p≤1 but fails for p>1, explaining successes and failures of this approach.

Contribution

It provides theoretical conditions under which l_p minimization can or cannot accurately recover multiple subspaces from data.

Findings

l_p minimization with 0<p≤1 can precisely recover subspaces

l_p minimization with p>1 cannot reliably recover subspaces

the results explain the success and failure cases of l_p energy minimization

Abstract

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 l_p-averaged distances of the sampled data points from any K subspaces. Under some conditions, we show that if $0<p\leq1$, then all underlying subspaces can be precisely recovered by l_p 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 l_p minimization. The results of this paper partially explain the successes and failures of the basic approach of l_p energy minimization for modeling data by multiple subspaces.