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Nearest Neighbor Searching Under
UncertaintyWuzhou Zhang
Supervised by Pankaj K. AgarwalDepartment of Computer Science
Duke University
Nearest Neighbor Searching (NNS)
Applications
Pattern Recognition, Data Compression
Statistical Classification, Clustering
Databases, Information Retrieval
Computer Vision, etc.http://en.wikipedia.org/wiki/Nearest_neighbor_search
Nearest Neighbor Searching Under Uncertainty
Discrete pdf•
Continuous pdf•
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Bisector In Case Of Gaussian
For Gaussian distribution, bisector is a line!
Hard to get explicit formula!
Figure: http://www.cs.utah.edu/~hal/courses/2009S_AI/Walkthrough/KalmanFilters/
In case of discrete pdf,
bisector is also a line!
In both cases, compute the Voronoi diagram, solve it optimally!
However, not a metric !
Squared Distance Function
bisector is simple and beautiful!
Sampling Continuous Distributions
Sometimes working on continuous distributions is hard….
Lower bounds on other metrics and distributions are also possible…. Let’s focus on discrete pdf then….
Expected Nearest NeighborIn L1 Metric ( cont. )
Source: Range Searching on Uncertain Data [P.K.Agarwal et al. 2009]
Building Block: Half-Space Intersection and Convex Hulls
Upper hulls correspond to lower envelopes, an example in 2D
Source: page 252 – 253, Computational Geometry: Algorithms and Applications, 3rd Edition[Mark de Berg et al. ]
Segment-tree Based Data Structures for Expected-NN In L1 Metric ( cont. )
Size of data structure
Preprocessing time
Query timeSummary of the result
Work harder in the near
future!
• Approximate the expected NN in L2 metric
• Study the complexity of expected Voronoi diagram
• Study the probability case
Future Work