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Near(est) Neighbor in High Dimensions

Alexandr Andoni (slides by Piotr Indyk). Near(est) Neighbor in High Dimensions. Nearest Neighbor. Given: A set P of points in R d Goal: build data structure which, for any query q, returns a point p  P minimizing ||p-q||. q. Solution for d=2 (sketch). Compute Voronoi diagram

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Near(est) Neighbor in High Dimensions

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  1. Alexandr Andoni (slides by Piotr Indyk) Near(est) Neighbor in High Dimensions

  2. Nearest Neighbor • Given: • A set P of points in Rd • Goal: build data structure which, for any query q, returns a point pP minimizing ||p-q|| q

  3. Solution for d=2 (sketch) • Compute Voronoi diagram • Given q, perform point location • Performance: • Space: O(n) • Query time: O(log n) (see 6.838 for details)

  4. NN in Rd • Exact algorithms use • Either nO(d) space, • Or O(dn) time • Approximate algorithms: • Space/time exponential in d[Arya-Mount-et al], [Kleinberg’97], [Har-Peled’02] • Space/time polynomial in d[Kushilevitz-Ostrovsky-Rabani’98], [Indyk-Motwani’98], [Indyk’98],…

  5. Eigenfaces 0.01 - 0.2 + 0.03 +……….. = = 0.02 + 0.03 +0.01 +……….. ?

  6. (Approximate) Near Neighbor • Near neighbor: • Given: • A set P of points in Rd, r>0 • Goal: build data structure which, for any query q, returns a point pP, ||p-q|| ≤ r (if it exists) • c-Approximate Near Neighbor: • Goal: build data structure which, for any query q: • If there is a point pP, ||p-q|| ≤ r • it returns p’P, ||p-q|| ≤ cr r q cr

  7. Locality-Sensitive Hashing [Indyk-Motwani’98] q • Idea: construct hash functions g: Rd U such that for any points p,q: • If ||p-q|| ≤ r, then Pr[g(p)=g(q)] is “high” • If ||p-q|| >cr, then Pr[g(p)=g(q)] is “small” • Then we can solve the problem by hashing p • “not-so-small”

  8. LSH • A family H of functions h: Rd → U is called (P1,P2,r,cr)-sensitive, if for any p,q: • if ||p-q|| <r then Pr[ h(p)=h(q) ] > P1 • if ||p-q|| >cr then Pr[ h(p)=h(q) ] < P2 • We hash using g(p)=h1(p).h2(p)…hk(p) • Intuition: amplify the probability gap • We can solve a c-approximate NN with: • Number of hash functions: n,  =log1/P2(1/P1) • Query time: O(d nlog n) • Space: O(n+1 +dn)

  9. LSH for Hamming metric [IM’98] • Functions: h(p)=pi, i.e., the i-th bit of p • We have Pr[ h(p)=h(q) ] = 1-D(p,q)/d • This gives exponent =1/c • Used for genetic sequence retrieval, motif finding, video sequence retrieval, compression, web clustering, pose estimation, etc.

  10. Other norms • Can embed l1d with coordinates in {1…M} into dM-dimensional Hamming space

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