Efficient Point-to-Subspace Query in $\ell^1$: Theory and Applications in Computer Vision

TL;DR

This paper introduces a method using Cauchy random embedding to efficiently identify the nearest subspace to a query point in high-dimensional space under  distance, with applications in robust vision tasks.

Contribution

It provides a theoretical foundation for using Cauchy embeddings to preserve nearest subspace relationships in  space, enabling faster search in high-dimensional vision applications.

Findings

Cauchy embedding preserves nearest subspace identity with constant probability

The approach enables efficient candidate selection for accurate search

Preliminary experiments support the theoretical claims in face and digit recognition

Abstract

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.