## 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 ℓ1 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 database subspaces into lower-dimensional space by random Cauchy matrix and solving small-scale distance evaluations (linear programs) in the projection space to locate the nearest candidates; (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 a low-order polynomial of the subspace dimension multiplied by a logarithm of the number of the subspaces (Theorem 2.1). The reduced dimensionality and hence complexity render the proposed algorithm particularly relevant to vision applications such as robust face and object instance recognition that we investigate empirically.

Original language | English (US) |
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Pages (from-to) | 2105-2138 |

Number of pages | 34 |

Journal | SIAM Journal on Imaging Sciences |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - Nov 6 2014 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2014 Society for Industrial and Applied Mathematics.

## Keywords

- Cauchy projection
- Face recognition
- Nearest subspace search
- Subspace modeling
- ℓ point-to-subspace distance

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