2026-09-19T13:52:08Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/333462022-01-13T07:54:29Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Piotr Indyk.
Samuel, Javed K. K
Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2006-07-13T15:17:04Z
2006-07-13T15:17:04Z
2005
2005
http://hdl.handle.net/1721.1/33346
62411992
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.
Includes bibliographical references (p. 71-73).
This thesis presents a lower bounds for embedding the Earth Mover Distance (EMID) metric into normed spaces. The EMID is a metric over two distributions where one is a mass of earth spread out in space and the other is a collection of holes in that same space. The EMD between these two distributions is defined as the least amount of work needed to fill the holes with earth. The EMD metric is used in a number of applications, for example in similarity searching and for image retrieval. We present a simple construction of point sets in the ENID metric space over two dimensions that cannot be embedded from the ED metric exactly into normed spaces, namely l1 and the square of l2. An embedding is a mapping f : X --> V with X a set of points in a metric space and ' Va set of points in some normed vector space. When the Manhattan distance is used as the underlying metric for the EMD, it can be shown that this example is isometric to K2,4 which has distortion equal to 1.25 when it is embedded into I and( 1.1180 when embedded into the square of 12. Other constructions of points sets in the EMID metric space over three and higher dimensisions are also discussed..
by Javed K.K. Samuel.
M.Eng.
73 p.
1891353 bytes
1894338 bytes
application/pdf
application/pdf
eng
Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Electrical Engineering and Computer Science.
Lower bounds for embedding the Earth Mover Distance metric into normed spaces
Thesis
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<Abstract>This thesis presents a lower bounds for embedding the Earth Mover Distance (EMID) metric into normed spaces. The EMID is a metric over two distributions where one is a mass of earth spread out in space and the other is a collection of holes in that same space. The EMD between these two distributions is defined as the least amount of work needed to fill the holes with earth. The EMD metric is used in a number of applications, for example in similarity searching and for image retrieval. We present a simple construction of point sets in the ENID metric space over two dimensions that cannot be embedded from the ED metric exactly into normed spaces, namely l1 and the square of l2. An embedding is a mapping f : X --> V with X a set of points in a metric space and ' Va set of points in some normed vector space. When the Manhattan distance is used as the underlying metric for the EMD, it can be shown that this example is isometric to K2,4 which has distortion equal to 1.25 when it is embedded into I and( 1.1180 when embedded into the square of 12. Other constructions of points sets in the EMID metric space over three and higher dimensisions are also discussed..</Abstract>
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