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SIMILARITY SEARCH The Metric Space Approach Pavel Zezula, Giuseppe Amato, Vlastislav Dohnal, Michal Batko Table of Contents Part I: Metric searching in a nutshell Foundations of metric space searching Survey of exiting approaches Part II: Metric searching in large collections Centralized index structures Approximate similarity search Parallel and distributed indexes P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 2 Approximate similarity search Approximate similarity search overcomes problems of exact similarity search using traditional access methods Similarity search returns mathematically precise result sets Moderate improvement of performance with respect to sequential scan Dimensionality curse Similarity is subjective so, in some cases, also approximate result sets satisfy the user Approximate similarity search processes query faster at the price of imprecision in the returned result sets Useful for instance in interactive systems Similarity search is an iterative process where temporary results are used to create a new query Improvements up to two orders of magnitude P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 3 Approximate similarity search Approximation strategies Relaxed pruning conditions Data regions overlapping the query regions can be discarded depending on the specific strategy Early termination of the search algorithm Search algorithm might stop before all regions have been accessed P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 4 Approximate Similarity Search relative error approximation (pruning condition) 1. 2. 3. 4. 5. 6. Range and k-NN search queries good fraction approximation small chance improvement approximation proximity-based approximation PAC nearest neighbor searching performance trials P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 5 Relative error approximation Let oN be the nearest neighbour of q. If d oA, q 1 e N d o ,q then oA is the (1+e)-approximate nearest neighbor of q This can be generalized to the k-th nearest neighbor d okA , q 1 e N d ok , q P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 6 Relative error approximation Exact pruning strategy: d q, p rq rp rp p rq q P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 7 Relative error approximation Approximate pruning strategy: rq d q, p rp rp 1 e p rq q rq/(1+e P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 8 Approximate Similarity Search relative error approximation (pruning condition) 1. good fraction approximation (stop condition) 2. 3. 4. 5. 6. Range and k-NN search queries K-NN search queries small chance improvement approximation proximity-based approximation PAC nearest neighbor searching performance trials P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 9 Good fraction approximation The k-NN algorithm determines the final result by reducing distances of current result set When the current result set belongs to a specific fraction of the objects closest to the query, the approximate algorithm stops Example: Stop when current result set belongs to the 10% of the objects closest to the query P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 10 Good fraction approximation For this strategy we use the distance distribution defined as Fq ( x) Prd (o, q) x The distance distribution Fq(x) specifies what is the probability that the distance of a random object o from q is smaller than x It is easy to see that Fq (x) gives, in probabilistic terms, the fraction of the database corresponding to the set of objects whose distance from q is smaller than x P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 11 Good fraction approximation 1 0,9 Fraction of the data set whose distances from q are smaller than d(q,ok) 0,8 0,7 0,6 0,5 0,4 Fq(x) d(q,ok) 0,3 ok 0,2 0,1 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 q P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 12 Good fraction approximation When Fq(d(ok,q)) < r all objects of the current result set belong to the fraction r of the dataset ok q P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 13 Good fraction approximation Fq(x) is difficult to be handled since we need to compute it for all possible queries It was proven that the overall distance distribution F(x) defined as follows F ( x) Prd (o1 , o2 ) x can be used in practice, instead of Fq(x), since they have statistically the same behaviour. F(x) can be easily estimated as a discrete function and it can be easily maintained in main memory P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 14 Approximate Similarity Search relative error approximation (pruning condition) 1. good fraction approximation (stop condition) 2. 5. 6. K-NN search queries small chance improvement approximation (stop c.) 3. 4. Range and k-NN search queries K-NN search queries proximity-based approximation PAC nearest neighbor searching performance trials P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 15 Small chance improvement approximation The M-Tree’s k-NN algorithm determines the final result by improving the current result set Each step of the algorithm the temporary result is improved and the distance of the k-th element decreases When the improvement of the temporary result set slows down, the algorithms can stop P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 16 Small chance improvement approximation f ( x) : d (q, o ) A k 0,38 0,36 0,34 Distance 0,32 0,3 0,28 0,26 0,24 0,22 0,2 0 500 1000 1500 Iteration P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 17 Small chance improvement approximation Function f (x) is not known a priori. A regression curve j (x), which approximate f (x), is computed using the least square method while the algorithm proceeds Through the derivative of j (x) it is possible to decide when the algorithm has to stop P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 18 Small chance improvement approximation The regression curve has the following form j ( x) c1j1 ( x) c2 where c1 and c2 are such that j c j (i) c i 0 1 1 2 f (i) 2 is minimum We have used both j1(x)=ln(x) and j1(x)=1/x P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 19 Regression curves 0,4 0,38 0,36 Distance 0,34 0,32 0,3 Distance Hyperbolic Regr. Logarithmic Regr. 0,28 0,26 0,24 0,22 0,2 0 500 1000 1500 Iteration P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 20 Approximate Similarity Search relative error approximation (pruning condition) 1. good fraction approximation (stop condition) 2. K-NN search queries proximity-based approximation (pruning cond.) 4. 6. K-NN search queries small chance improvement approximation (stop c.) 3. 5. Range and k-NN search queries Range and k-NN search queries PAC nearest neighbor searching performance trials P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 21 Proximity-based approximation Regions whose probability of containing qualifying objects is below a certain threshold are pruned even if they overlap the query region Proximity between regions is defined as the probability that a randomly chosen object appears in both the regions. This resulted in an increase of performance of two orders of magnitude both for range queries and nearest neighbour queries P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 22 Proximity-based approximation R1.11 1.R 1.q 1.R 1R.2 R 1.R 3 1.3 P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 23 Approximate Similarity Search relative error approximation (pruning condition) 1. good fraction approximation (stop condition) 2. K-NN search queries small chance improvement approximation (stop c.) 3. K-NN search queries proximity-based approximation (pruning cond.) 4. Range and k-NN search queries PAC nearest neighbor searching (pruning & stop) 5. 6. Range and k-NN search queries 1-NN search queries performance trials P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 24 PAC nearest neighbour searching It uses the same time a relaxed branching condition and a stop condition The relaxed branching condition is the same used for the relative error approximation to find an (1+e)-approximate-nearest neighbor In addition it halts prematurely when the probability that we have found the (1+e)-approximate-nearest neighbor is above the threshold d It can only be used for 1-NN search queries P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 25 PAC nearest neighbour searching Let us suppose that then nearest neighbour found so far is oA Let eact be the actual error on distance of oA d oA, q e act 1 N d o ,q The algorithm stops if Pre act e d The above probability is obtained by computing the distribution of the distance of the nearest neighbor. P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 26 PAC nearest neighbour searching Distribution of the distance of the nearest neighbor in X (of cardinality n) with respect to q: Gq ( x) Pro X : d (q, o) x 1 (1 Fq ( x))n Given that P re act e P r o X : d (q, o A ) / d (q, o) 1 e P r o X : d (q, o) d (q, o A ) /(1 e ) Gq d (q, o A ) /(1 e ) The algorithm halts when Gq d (q, o A ) /(1 e ) d P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 27 Approximate Similarity Search relative error approximation (pruning condition) 1. good fraction approximation (stop condition) 2. K-NN search queries small chance improvement approximation (stop c.) 3. K-NN search queries proximity-based approximation (pruning cond.) 4. Range and k-NN search queries PAC nearest neighbor searching (pruning & stop) 5. 6. Range and k-NN search queries 1-NN search queries performance trials P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 28 Comparisons tests Tests on a dataset of 11,000 objects Objects are vectors of 45 dimensions We compared the five approximation approaches Range queries tested on the methods: Relative error Proximity Nearest-neighbors queries tested on all methods P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 29 Comparisons: range queries IE Relative error 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 r=1,800 r=2,200 r=2,600 r=3,000 0 0.2 0.4 0.6 0.8 1 R P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 30 Comparisons: range queries Proximity 7 6 IE 5 r=1,800 r=2,200 r=2,600 r=3,000 4 3 2 1 0 0.2 0.4 0.6 0.8 1 R P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 31 Comparisons NN queries Relative error 1.6 1.5 IE 1.4 k=1 k=3 k=10 k=50 1.3 1.2 1.1 1 0 0.001 0.002 0.003 0.004 EP P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 32 Comparisons NN queries Good fraction 800 700 600 k=1 k=3 k=10 k=50 IE 500 400 300 200 100 0 0 0.01 0.02 0.03 EP P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 33 Comparisons NN queries IE Small chance improvement 200 180 160 140 120 100 80 60 40 20 0 k=1 k=3 k=10 k=50 0 0.02 0.04 0.06 0.08 0.1 EP P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 34 Comparisons NN queries Proximity 800 700 600 k=1 k=3 k=10 k=50 IE 500 400 300 200 100 0 0 0.005 0.01 0.015 0.02 0.025 0.03 EP P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 35 Comparisons NN queries IE PAC 500 450 400 350 300 250 200 150 100 50 0 eps=2 eps=3 eps=4 0 0.001 0.002 0.003 0.004 0.005 EP P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 36 Conclusions: Approximate similarity search in metric spaces These techniques for approximate similarity search can be applied to generic metric spaces Vector spaces are a special case of metric space. High accuracy of approximate results are generally obtained with high improvement of efficiency Best performance obtained with the good fraction approximation methods The proximity based is a bit worse than good fraction approximation but can be used for range queries and k-NN queries. P. Zezula, G. Amato, V. Dohnal, M. Batko: Similarity Search: The Metric Space Approach Part I, Chapter 1 37