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Nearest Neighbor Searching Under Uncertainty: Methods and Guarantees

Explore methods for Exact Nearest Neighbor (ENN) queries with performance guarantees under uncertain data scenarios. Discover how uncertainty impacts proximity in sensor databases, face recognition, and mobile data. Future work includes linear-size index for most likely NN queries.

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Nearest Neighbor Searching Under Uncertainty: Methods and Guarantees

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  1. Joint work with Pankaj K. Agarwal, AlonEfrat, and SwaminathanSankararaman Nearest-Neighbor Searching Under UncertaintyWuzhou ZhangDepartment of Computer Science, Duke University PODS, May 23, 2012

  2. Nearest-Neighbor Searching a set of points in any query point in Find the closest point to

  3. Voronoi Diagram • Voronoi cell: • Voronoidiagram: decomposition induced by

  4. Data Uncertainty • Location of data is imprecise: Sensor databases, face recognition, mobile data, etc. What is the “nearest neighbor” of now?

  5. Our Model and Problem Statement • Uncertain point : represented as a probability density function(pdf) -- • Expected distance: • . Find the expected nearest neighbor (ENN) of : Or an -ENN:

  6. Previous work • Uncertain data • ENN • The ENN under metric: ε-approximation [Ljosa2007] • No bounds on the running time • Most likely NN • Heuristics [Cheng2008, Kriegel2007, Cheng2004, etc] • Uncertain query • ENN • Discrete uniform distribution: both exact and O(1) factor approximation [Li2011, Sharifzadeh2010, etc] • No bounds on the running time

  7. Our contribution Firstnontrivial methods for ENN queries with provable performance guarantees ! Results in , extends to higher dimensions

  8. Expected Voronoi Diagram • Expected Voronoi cell • Expected Voronoi diagram : induced by • An example in metric

  9. Squared Euclidean distanceUncertain data • : the centroid of • Lemma: • same as the weighted Voronoi diagram WVD Remarks: Works for any distribution

  10. metricUncertain data • Size of : • Lower bound construction the inverse Ackermann function Remarks: Extends to metric

  11. metricUncertain data (cont.) • A near-linear size index exists despite size of Remarks: Extends to higher dimensions

  12. Euclidean metric (-ENN)Uncertain data • Approximateby • Outside the grid: • Inside the gird: • Total # of cells: Cell size: 𝜀 Remarks: Extends to any metric

  13. Euclidean metric (-ENN)Uncertain data (cont.) • A linear size approximate ! 13

  14. Conclusion and future work • Conclusion • Firstnontrivial methods for answering exact or approximate ENN queries with provable performance guarantees • ENN is not a good indicator when the variance is large • Future work • Linear-size index for most likely NN queries in sublineartime • Index for returning the probability distribution of NNs Thanks

  15. Squared Euclidean distanceUncertain query • : the centroid of • Preprocessing • Compute the Voronoi diagram VD • Query • Given , compute in , then query VD with Remarks: Extends to higher dimensions and works for any distribution

  16. Rectilinear metricUncertain query • Similarly, linear pieces

  17. Euclidean metric (-ENN)Uncertain query Remarks: Extends to higher dimensions

  18. metricUncertain data (cont.) • A near-linear size index exists despite size of • linear pieces! Linear!

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