approximate nearest neighbors: towards removing the curse of dimensionality
DESCRIPTION
Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality. Piotr Indyk, Rajeev Motwani. The 30 th annual ACM symposium on theory of computing 1998. Problems. Nearest neighbor (NN) problem: - PowerPoint PPT PresentationTRANSCRIPT
Approximate Nearest Neighbors: Towards Removing the Curse of
Dimensionality
Piotr Indyk, Rajeev Motwani
The 30th annual ACM symposium on theory of computing 1998
Problems
• Nearest neighbor (NN) problem:– Given a set of n points P={p1, …, pn} in some metric sp
ace X, preprocess P so as to efficiently answer queries which require finding the point in P closest to a query point qX.
• Approximate nearest neighbor (ANN) problem:– Find a point pP that is an –approximate nearest nei
ghbor of the query q in that for all p'P, d(p,q)(1+)d(p',q).
Motivation
• The nearest neighbors problem is of major importance to a variety of applications, usually involving similarity searching. – Data compression– Databases and data mining– Information retrieval– Image and video databases– Machine learning – Pattern recognition – Statistics and data analysis
• Curse of dimensionality– The curse of dimensionality is a term coined by Richard
Bellman to describe the problem caused by the exponential increase in volume associated with adding extra dimensions to a (mathematical) space.
Overview of results and techniques
• These results are obtained by reducing -NNS to a new problem: point location in equal balls.
nearest neighbor search (NNS)-nearest neighbor search (NNS)
Ring-Cover Trees
Point location in equal balls (PLEB)- Point location in equal balls (PLEB)
Locality-Sensitive Hashing
Proposition 1 Proposition 2
The Bucketing method
Proposition 3
Random projections
Content
The Bucketing method
• We decompose each ball into a bounded number of cells and store them in a dictionary.
• The bucketing algorithm works for any lp norm.