Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality

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Approximate Nearest Neighbors Towards Removing
the Curse of Dimensionality

• Piotr Indyk, Rajeev Motwani

The 30th annual ACM symposium on theory of
computing 1998
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Problems

• Nearest neighbor (NN) problem
• Given a set of n points Pp1, , pn in some
  metric space X, preprocess P so as to efficiently
  answer queries which require finding the point in
  P closest to a query point q?X.
• Approximate nearest neighbor (ANN) problem
• Find a point p?P that is an ?approximate nearest
  neighbor of the query q in that for all p'?P,
  d(p,q)?(1?)d(p',q).

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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.

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Overview of results and techniques

• These results are obtained by reducing ?-NNS to a
  new problem point location in equal balls.

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Content
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Definitions
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Theorems
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Constructing Ring-cover trees
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Analysis of Ring-cover trees
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Definitions
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Locality-Sensitive Hashing
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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.

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J. L. Lemma