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Similarity Search on Bregman Divergence, Towards Non- Metric Indexing Zhenjie Zhang, Beng Chi Ooi, Srinivasan Parthasarathy, Anthony K. H. Tung

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Metric v.s. Non-Metric Euclidean distance dominates DB queries Similarity in human perception Metric distance is not enough! Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 2

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Outline Bregman Divergence Solution Basic solution Better pruning bounds Query distribution Experiments Conclusion Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 3

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Bregman Divergence Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 4 qp Euclidean dist. convex function f(x) Bregman divergence D f (p,q) (q,f(q)) (p,f(p)) h

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Bregman Divergence Mathematical Interpretation The distance between p and q is defined as the difference between f(p) and the first order Taylor expansion at q Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 5 original f(x)first order Taylor expansion of f(x) at q

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Bregman Divergence General Properties Uniqueness A function f(x) uniquely decides the D f (p,q) Non-Negativity D f (p,q)≥0 for any p, q Identity D f (p,p)=0 for any p Symmetry and Triangle Inequality Do NOT hold any more Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 6

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Examples Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 7 Distancef(x)Df(p,q)Df(p,q)Usage KL-Divergencex logxp log (p/q)distribution, color histogram Itakura-Saito Distance -logxp/q-log (p/q)-1signal, speech Squared Euclidean x2x2 (p-q)2(p-q)2 traditional queries Von-Nuemann Entropy tr(X log X – X)tr(X logX – X logY – X + Y) symmetric matrix

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Why in DB system? Database application Retrieval of similar images, speech signals, or time series Optimization on matrices in machine learning Efficiency is important! Query Types Nearest Neighbor Query Range Query Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 8

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Euclidean Space How to answer the queries R-Tree Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 9

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Euclidean Space How to answer the queries VA File Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 10

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Our goal Re-use the infrastructure of existing DB system to support Bregman divergence Storage management Indexing structures Query processing algorithms Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 11

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Outline Bregman Divergence Solution Basic solution Better pruning bounds Query distribution Experiments Conclusion Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 12

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Basic Solution Extended Space Convex function f(x) = x Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 13 pointA1A1 A2A2 p01 q0.5 r10.8 t pointA1A1 A2A2 A3A3 p+p+ 011 q+q+ 0.5 r+r t+t

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Basic Solution After the extension Index extended points with R-Tree or VA File Re-use existing algorithms with new lower and upper bound computation Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 14

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How to improve? Reformulation of Bregman divergence Tighter bounds are derived No change on index construction or query processing algorithm Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 15

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A New Formulation Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 16 qp D f (p,q)+Δ query vector v q D* f (p,q) h h’

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Math. Interpretation Reformulation of similarity search queries k-NN query: query q, data set P, divergence D f Find the point p, minimizing Range query: query q, threshold θ, data set P Return any point p that Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 17

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Naïve Bounds Check the corners of the bounding rectangles Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 18

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Tighter Bounds Take the curve f(x) into consideration Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 19

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Query distribution Distortion of rectangles The difference between maximum and minimum distances from inside the rectangle to the query Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 20

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Can we improve it more? When Building R-Tree in Euclidean space Minimize the volume/edge length of MBRs Does it remain valid? Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 21

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Query distribution Distortion of bounding rectangles Invariant in Euclidean space (triangle inequality) Query-dependent for Bregman Divergence Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 22

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Utilize Query Distribution Summarize query distribution with O(d) real number Estimation on expected distortion on any bounding rectangle in O(d) time Allows better index to be constructed for both R-Tree and VA File Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 23

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Outline Bregman Divergence Solution Basic solution Better pruning bounds Query distribution Experiments Conclusion Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 24

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Experiments Data Sets KDD’99 data Network data, the proportion of packages in 72 different TCP/IP connection Types DBLP data Use co-authorship graph to generate the probabilities of the authors related to 8 different areas Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 25

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Experiment Data Sets Uniform Synthetic data Generate synthetic data with uniform distribution Clustered Synthetic data Generate synthetic data with Gaussian Mixture Model Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 26

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Experiments Methods to compare Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 27 BasicImproved Bounds Query Distribution R-TreeRR-BR-BQ VA FileVV-BV-BQ Linear ScanLS BB-TreeBBT

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Experiments Index Construction Time Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 28

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Experiments Varying dimensionality Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 29

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Experiments Varying dimensionality (cont.) Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 30

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Experiments Varying k for nearest neighbor query Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 31

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Conclusion A general technique on similarity for Bregman Divergence All techniques are based on existing infrastructure of commercial database Extensive experiments to compare performances with R-Tree and VA File with different optimizations Similarity Search on Bregman Divergence: Towards Non-Metric Indexing 32

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Acknowledgment Zhenjie Zhang, Anthony K. H. Tung and Beng Chin Ooi were supported by Singapore NRF grant R Srinivasan Parthasarathy was supported by NSF IIS (CAREER) and NSF CCF

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Q & A

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