This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Dimensionality Reduction and Similarity Computation by Inner-Product Approximations
June 2004 (vol. 16 no. 6)
pp. 714-726

Abstract—As databases increasingly integrate different types of information such as multimedia, spatial, time-series, and scientific data, it becomes necessary to support efficient retrieval of multidimensional data. Both the dimensionality and the amount of data that needs to be processed are increasing rapidly. Reducing the dimension of the feature vectors to enhance the performance of the underlying technique is a popular solution to the infamous curse of dimensionality. We expect the techniques to have good quality of distance measures when the similarity distance between two feature vectors is approximated by some notion of distance between two lower-dimensional transformed vectors. Thus, it is desirable to develop techniques resulting in accurate approximations to the original similarity distance. In this paper, we investigate dimensionality reduction techniques that directly target minimizing the errors made in the approximations. In particular, we develop dynamic techniques for efficient and accurate approximation of similarity evaluations between high-dimensional vectors based on inner-product approximations. Inner-product, by itself, is used as a distance measure in a wide area of applications such as document databases. A first order approximation to the inner-product is obtained from the Cauchy-Schwarz inequality. We extend this idea to higher order power symmetric functions of the multidimensional points. We show how to compute fixed coefficients that work as universal weights based on the moments of the probability density function of the data set. We also develop a dynamic model to compute the universal coefficients for data sets whose distribution is not known. Our experiments on synthetic and real data sets show that the similarity between two objects in high-dimensional space can be accurately approximated by a significantly lower-dimensional representation.

[1] A. Acharya, M. Uysal, and J. Saltz, Active Disks: Programming Model, Algorithms and Evaluation Proc. Eighth Int'l Conf. Architectural Support for Programming Languages and Operating Systems, pp. 81-91, May 1998.
[2] R. Agrawal, C. Faloutsos, and A. Swami, Efficient Similarity Search in Sequence Databases Proc. Fourth Int'l Conf. Foundations of Data Organization and Algorithms, pp. 69-84, 1993.
[3] N. Beckmann, H. Kriegel, R. Schneider, and B. Seeger, The R*Tree: An Efficient and Robust Access Method for Points and Rectangles Proc. ACM SIGMOD Int'l Conf. Management of Data, pp. 322-331, May 1990.
[4] S. Berchtold, C. Bohm, D. Keim, and H. Kriegel, A Cost Model for Nearest Neighbor Search in High-Dimensional Data Space Proc. ACM Symp. Principles of Database Systems, 1997.
[5] S. Berchtold, C. Bohm, and H.-P. Kriegel, The Pyramid-Technique: Towards Breaking the Curse of Dimensionality Proc. ACM SIGMOD Int'l Conf. Management of Data, pp. 142-153, June 1998.
[6] S. Berchtold, D. Keim, and H. Kriegel, The X-Tree: An Index Structure for High-Dimensional Data Proc. Int'l Conf. Very Large Data Bases, pp. 28-39, 1996.
[7] P. Bernstein, M. Brodie, S. Ceri, D. DeWitt, M. Franklin, H. Garcia-Molina, J. Gray, J. Held, J. Hellerstein, H. Jagadish, M. Lesk, D. Maier, J. Naughton, H. Pirahesh, M. Stonebraker, and J. Ullman, The Asilomar Report on Database Research Sigmod Record, vol. 27, no. 4, Dec. 1998.
[8] E. Bingham, H. Mannila, Random Projection in Dimensionality Reduction: Applications to Image and Text Data Proc. Int'l Conf. Knowledge Discovery and Data Mining, 2001.
[9] K.R. Castleman, Digital Image Processing. Prentice-Hall 1996.
[10] K. Chakrabarti and S. Mehrotra, Local Dimensionality Reduction: A New Approach to Indexing High Dimensional Spaces The VLDB J., pp. 89-100, 2000.
[11] M.S. Charikar, Similarity Estimation Techniques from Rounding Algorithms Proc. 34th Ann. ACM Symp. Theory of Computing, 2002.
[12] X. Cheng, R. Dolin, M. Neary, S. Prabhakar, K. Ravikanth, D. Wu, D. Agrawal, A. El Abbadi, M. Freeston, A. Singh, T. Smith, and J. Su, Scalable Access within the Context of Digital Libraries IEEE Proc. Int'l Conf. Advances in Digital Libraries (ADL), pp. 70-81, 1997.
[13] S. Dasgupta and A. Gupta, An Elementary Proof of the Johnson-Lindenstrauss Lemma Technical Report TR-99-006, Int'l Computer Science Inst., Berkeley, 1999.
[14] S. Deerwester, S.T. Dumais, G.W. Furnas, T.K. Launder, and R. Harshman, Indexing by Latent Semantic Analysis J. Am. Soc. for Information Science, vol. 41, pp. 391-407, 1990.
[15] D. Hull, Improving Text Retrieval for the Routing Problem Using Latent Semantic Indexing Proc. 17th ACM-SIGIR Conf., pp. 282-291, 1994.
[16] S.T. Dumais, Improving the Retrieval of Information from External Sources Behavior Research Methods, Instruments and Computers, vol. 23, pp. 229-236, 1991.
[17] Ö. Egecioglu, How to Approximate the Inner-Product: Fast Dynamic Algorithms for Similarity Technical Report TRCS98-37, Dept. of Computer Science, Univ. of California at Santa Barbara, Dec. 1998.
[18] Ö. Egecioglu and H. Ferhatosmanoglu, Dimensionality Reduction and Similarity Distance Computation by Inner Product Approximations Proc. Ninth ACM Int'l Conf. Information and Knowledge Management, pp. 219-226, Nov. 2000.
[19] C. Faloutsos, R. Barber, M. Flickner, J. Hafner, W. Niblack, D. Petkovic, and W. Equitz, Efficient and Effective Querying by Image Content J. Intelligent Information Systems, vol. 3, pp. 231-262, 1994.
[20] C. Faloutsos, M. Ranganathan, and Y. Manolopoulos, Fast Subsequence Matching in Time-Series Databases Proc. ACM SIGMOD Int'l Conf. Management of Data, pp. 419-429, May 1994.
[21] H. Ferhatosmanoglu, E. Tuncel, D. Agrawal, and A. El Abbadi, Vector Approximation Based Indexing for Non-Uniform High Dimensional Data Sets Proc. Ninth ACM Int'l Conf. Information and Knowledge Management, pp. 202-209, Nov. 2000.
[22] H. Ferhatosmanoglu, E. Tuncel, D. Agrawal, and A. El Abbadi, Approximate Nearest Neighbor Searching in Multimedia Databases Proc. 17th IEEE Int'l Conf. Data Eng. (ICDE), pp. 503-511, Apr. 2001.
[23] A. Gionis, P. Indyk, and R. Motwani, Similarity Searching in High Dimensions via Hashing Proc. Int'l Conf. Very Large Data Bases, pp. 518-529, Sept. 1999.
[24] A. Guttman, R-Trees: A Dynamic Index Structure for Spatial Searching Proc. ACM SIGMOD Int'l Conf. Management of Data, pp. 47-57, 1984.
[25] N.A.J. Hastings and J.B. Peacock, Statistical Distributions. New York, Halsted Press, 1975.
[26] N.S. Jayant and P. Noll, Digital Coding of Waveforms. Prentice-Hall, 1984.
[27] T. Kailath, Modern Signal Processing. Springer Verlag, 1985.
[28] K.V.R. Kanth, D. Agrawal, and A. Singh, Dimensionality Reduction for Similarity Searching in Dynamic Databases Proc. ACM SIGMOD Int'l Conf. Management of Data, 1998.
[29] F. Korn, N. Sidiropoulos, C. Faloutsos, E. Siegel, and Z. Protopapas, Fast Nearest Neighbor Search in Medical Image Databases Proc. Int'l Conf. Very Large Data Bases, pp. 215-226, 1996.
[30] K. Lin, H.V. Jagadish, and C. Faloutsos, The TV-Tree: An Index Structure for High-Dimensional Data VLDB J., vol. 3, pp. 517-542, 1995.
[31] D.B. Lomet and B. Salzberg, The hb-Tree: A Multi-Attribute Indexing Method with Good Guaranteed Performance ACM Trans. Database Systems, vol. 15, no. 4, pp. 625-658, Dec. 1990.
[32] B.S. Manjunath and W.Y. Ma, “Texture Features for Browsing and Retrieval of Image Data,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 8, pp. 837-842, Aug. 1996
[33] W. Niblack, R. Barber, W. Equitz, M. Flickner, E. Glasman, D. Petkovic, and P. Yanker, The QBIC Project: Querying Images by Content Using Color, Texture and Shape Proc. SPIE Conf. 1908 on Storage and Retrieval for Image and Video Databases, vol. 1908, pp. 173-187, Feb. 1993.
[34] A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing. Prentice-Hall, 1989.
[35] J.T. Robinson, The kdb-Tree: A Search Structure for Large Multi-Dimensional Dynamic Indexes Proc. ACM SIGMOD Int'l Conf. Management of Data, pp. 10-18, 1981.
[36] SciDAC, Scientific data management center,http://sdm.lbl.govsdmcenter/, 2002.
[37] T. Seidl and H.-P. Kriegel, Efficient User-Adaptable Similarity Search in Large Multimedia Databases Proc. Int'l Conf. Very Large Data Bases, pp. 506-515, 1997.
[38] T. Seidl and H.P. Kriegel, Optimal Multistep k-Nearest Neighbor Search Proc. ACM SIGMOD Int'l Conf. Management of Data, June 1998.
[39] V.S. Subrahmanian, Principles of Multimedia Database Systems. Morgan Kaufmann Publishers, 1999.
[40] M. Vlachos, C. Domeniconi, D. Gunopulos, G. Kollios, and N. Koudas, Non-Linear Dimensionality Reduction Techniques for Classification and Visualization Proc. Eighth ACM SIGKDD Int'l Conf. Knowledge Discovery and Data Mining, July 2002.
[41] D. White and R. Jain, “Similarity Indexing with the SS-Tree,” Proc. 12th Int'l Conf. Data Eng., 1996.
[42] D. Wu, D. Agrawal, A. El Abbadi, and T.R. Smith, Efficient Retrieval for Browsing Large Image Databases Proc. Conf. Information and Knowledge Management, pp. 11-18, Nov. 1996.
[43] Y. Wu, D. Agrawal, and A. El Abbadi, A Comparison of DFT and DWT Based Similarity Search in Time-Series Databases Proc. Ninth Int'l Conf. Information and Knowledge Management, 2000.

Index Terms:
Inner-product approximation, dimensionality reduction, p{\hbox{-}}\rm NORMS, similarity search, high-dimensional data.
Citation:
?mer Egecioglu, Hakan Ferhatosmanoglu, Umit Ogras, "Dimensionality Reduction and Similarity Computation by Inner-Product Approximations," IEEE Transactions on Knowledge and Data Engineering, vol. 16, no. 6, pp. 714-726, June 2004, doi:10.1109/TKDE.2004.9
Usage of this product signifies your acceptance of the Terms of Use.