  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 that Cauchy random embedding of the objects into
significantly-lower-dimensional spaces helps preserve the identity of the
nearest subspace with constant probability. This offers the possibility of
efficiently selecting several candidates for accurate search. We sketch
preliminary experiments on robust face and digit recognition to corroborate our
theory.
