Efficient Point-to-Subspace Query in $\ell^1$ with Application to Robust Face Recognition

Motivated by vision tasks such as robust face and object recognition, we consider the following general problem: given a collection of low-dimensional linear subspaces in a high-dimensional ambient (image) space, and a query point (image), efficiently determine the nearest subspace to the query in $\ell^1$ distance. We show in theory this problem can be solved with a simple two-stage algorithm: (1) random Cauchy projection of query and subspaces into low-dimensional spaces followed by efficient distance evaluation ($\ell^1$ regression); (2) getting back to the high-dimensional space with very few candidates and performing exhaustive search. We present preliminary experiments on robust face recognition to corroborate our theory.