1. Nearest Neighbor Retrieval Using Distance-Based Hashing Michalis Potamias and Panagiotis Papapetrou supervised by Prof George Kollios A method is proposed for indexing spaces with arbitrary distance measures, so as to achieve efficient approximate nearest neighbor retrieval. Hashing methods, such as Locality Sensitive Hashing (LSH), have been successfully applied for similarity indexing in vector spaces and string spaces under the Hamming distance. The key novelty of the hashing technique proposed here is that it can be applied to spaces with arbitrary distance measures. First, we describe a domain-independent method for constructing a family of binary hash functions. Then, we use these functions to construct multiple multi-bit hash tables. We show that the LSH formalism is not applicable for analyzing the behavior of these tables as index structures. We present a novel formulation, that uses statistical observations from sample data to analyze retrieval accuracy and efficiency for the proposed indexing method. Experiments on several real-world data sets demonstrate that our method produces good trade-offs between accuracy and efficiency, and significantly outperforms VP-trees, which are a well-known method for distance-based indexing. HierarchicalDBH Rank Queries according to D(Q,N(Q) Divide space into disjoint subsets (equi-height) Train separate indices for each subset Reduce Hash Cost Use small number of “pseudoline” points Database Group Problem NEAREST NEIGHBOR: Given a database S, a distance function D our task is: for a previous unseen query q, locate a point p of the database such that the distance between q and every point o of the database is greater or equal than the distance between p and q. COST MODEL: Minimize number of Distance Computations Computing D may be very expensive Dynamic Time Warping for Time Series Edit Distance Variants for DNA alignment PROBLEM DEFINITION: Define index structure to answer Nearest Neighbor queries efficiently A SOLUTION: Brute Force! Try them all and get the exact answer OUR SOLUTION: Are we willing to trade accuracy for efficiency ? Hash Based Indexing Idea: 1. Come up with hash functions that hash similar objects to similar buckets 2. Hash every database object to some buckets 3. At query time apply the same hash function to the query 4. Filter: Retrieve the collisions. The rest of the database is pruned. 5. Refine: Compute actual distances. Return the object with the smallest distance as the NN. Locality Sensitive Hashing Locality Sensitive Family of Functions Amplify the gap between p 1 and p 2 : Randomly pick l hash vectors of k functions each. Probability of collision in at least one of l hash tables: H using Pseudoline Projections ( H DBH ) Works on Arbitrary Space but is not Locality Sensitive! Define a line projection function that maps an arbitrary space into the real line R: Real valued  Discrete valued: Hash tables should be balanced. Thus t1, t2 are chosen from V: ACCURACY vs. EFFICIENCY: How often is the actual NN retrieved? How much time does NN retrieval take? Analysis  Probability of collision between any two objects:  Same probability on a k-bit hash table:  Prob of collision in at least one of the l hash tables:  Accuracy, i.e. the probability over all queries Q that we will retrieve the nearest neighbor N(Q):  LookupCost: Expected number of objects that collide in at least one of the l hash tables  HashCost: # of distance computations to evaluate h-functions:  Total Cost per query:  Efficiency (for all Queries): Use Sampling to estimate Accuracy and Efficiency 1. Sample Queries 2. Sample Database Objects 3. Sample Hash Functions 4. Compute Integrals Finding optimal k & l..given accuracy (say 90%)…..For k=1,2,…..compute smallest l that yields required accuracy. Typically, optimal k : last k for which efficiency improves. C(Q,N(Q)) Number of Queries Additional Optimizations Experiments D(Q,N(Q)) Conclusion General purpose Distance is black box Does not require metric properties Statistical analysis is possible Even when NN is not returned, a very close N is returned… For many applications that’s fine!! Not sublinear in size of DB Statistical (not probabilistic) Need “representative” sample sets Hands dataset.. actual performance was different than simulation.. – the training set was not representative!