Abstract
We develop linear sketches for estimating the Earth-Mover distance between two point sets, i.e., the cost of the minimum weight matching between the points according to some metric. While Euclidean distance and Edit distance are natural measures for vectors and strings respectively, Earth-Mover distance is a well-studied measure that is natural in the context of visual or metric data. Our work considers the case where the points are located at the nodes of an implicit graph and define the distance between two points as the length of the shortest path between these points. We first improve and simplify an existing result by Brody et al. [4] for the case where the graph is a cycle. We then generalize our results to arbitrary graph metrics. Our approach is to recast the problem of estimating Earth-Mover distance in terms of an ℓ1 regression problem. The resulting linear sketches also yield space-efficient data stream algorithms in the usual way.
Supported by NSF CAREER Award CCF-0953754 and associated REU Supplement.
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References
Ahn, K.J., Guha, S., McGregor, A.: Analyzing graph structure via linear measurements. In: SODA, pp. 459–467 (2012)
Ahn, K.J., Guha, S., McGregor, A.: Graph sketches: sparsification, spanners, and subgraphs. In: PODS, pp. 5–14 (2012)
Andoni, A., Ba, K.D., Indyk, P., Woodruff, D.P.: Efficient sketches for earth-mover distance, with applications. In: FOCS, pp. 324–330 (2009)
Brody, J., Liang, H., Sun, X.: Space-efficient approximation scheme for circular earth mover distance. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 97–108. Springer, Heidelberg (2012)
Cabrelli, C., Molter, U.: A linear time algorithm for a matching problem on the circle. Inf. Process. Lett. 66(3), 161–164 (1998)
Clarkson, K.L., Drineas, P., Magdon-Ismail, M., Mahoney, M.W., Meng, X., Woodruff, D.P.: The fast cauchy transform and faster robust linear regression. In: SODA, pp. 466–477 (2013)
Cormode, G., Garofalakis, M.N., Haas, P.J., Jermaine, C.: Synopses for massive data: Samples, histograms, wavelets, sketches. Foundations and Trends in Databases 4(1-3), 1–294 (2012)
Indyk, P.: A near linear time constant factor approximation for euclidean bichromatic matching (cost). In: SODA, pp. 39–42 (2007)
Indyk, P., McGregor, A., Newman, I., Onak, K. (eds.): Open Problems in Data Streams, Property Testing, and Related Topics (2011), http://www.cs.umass.edu/~mcgregor/papers/11-openproblems.pdf
Indyk, P., Price, E.: K-median clustering, model-based compressive sensing, and sparse recovery for earth mover distance. In: STOC, pp. 627–636 (2011)
Kane, D.M., Nelson, J., Porat, E., Woodruff, D.P.: Fast moment estimation in data streams in optimal space. In: STOC, pp. 745–754 (2011)
McGregor, A. (ed.): Open Problems in Data Streams and Related Topics (2007), http://www.cse.iitk.ac.in/users/sganguly/data-stream-probs.pdf
McGregor, A., Rudra, A., Uurtamo, S.: Polynomial fitting of data streams with applications to codeword testing. In: STACS, pp. 428–439 (2011)
Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision 40(2), 99–121 (2000)
Sohler, C., Woodruff, D.P.: Subspace embeddings for the l1-norm with applications. In: STOC, pp. 755–764 (2011)
Tirthapura, S., Woodruff, D.P.: Rectangle-efficient aggregation in spatial data streams. In: PODS, pp. 283–294 (2012)
Verbin, E., Zhang, Q.: Rademacher-sketch: A dimensionality-reducing embedding for sum-product norms, with an application to earth-mover distance. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 834–845. Springer, Heidelberg (2012)
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McGregor, A., Stubbs, D. (2013). Sketching Earth-Mover Distance on Graph Metrics. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_20
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DOI: https://doi.org/10.1007/978-3-642-40328-6_20
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