Lower bounds for embedding the Earth Mover Distance metric into normed spaces

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..