A randomized algorithm for closest-point queries
SIAM Journal on Computing
Point location in arrangements of hyperplanes
Information and Computation
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Efficient Search for Approximate Nearest Neighbor in High Dimensional Spaces
SIAM Journal on Computing
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Dimensionality Reductions in ℓ2 that Preserve Volumes and Distance to Affine Spaces
Discrete & Computational Geometry
Efficient point-to-subspace query in ℓ1 with application to robust face recognition
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part IV
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We consider the problem of approximate nearest neighbors in high dimensions, when the queries are lines. In this problem, given n points in Rd, we want to construct a data structure to support efficiently the following queries: given a line L, report the point p closest to L. This problem generalizes the more familiar nearest neighbor problem. From a practical perspective, lines, and low-dimensional flats in general, may model data under linear variation, such as physical objects under different lighting. For approximation 1 + ε, we achieve a query time of d3n0.5+t, for arbitrary small t 0, with a space of d2nO(1/ε2+1/t2). To the best of our knowledge, this is the first algorithm for this problem with polynomial space and sub-linear query time.