Efficient Point-to-Subspace Query in $\ell^1$ with Application to Robust Object Instance 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. In contrast to the naive exhaustive search which entails large-scale linear programs, we show that the computational burden can be cut down significantly by a simple two-stage algorithm: (1) projecting the query and data-base subspaces into lower-dimensional space by random Cauchy matrix, and solving small-scale distance evaluations (linear programs) in the projection space to locate candidate nearest; (2) with few candidates upon independent repetition of (1), getting back to the high-dimensional space and performing exhaustive search. To preserve the identity of the nearest subspace with nontrivial probability, the projection dimension typically is low-order polynomial of the subspace dimension multiplied by logarithm of number of the subspaces (Theorem 2.1). The reduced dimensionality and hence complexity renders the proposed algorithm particularly relevant to vision application such as robust face and object instance recognition that we investigate empirically.