Approximation Algorithms for Computing the Earth Mover's Distance Under Transformations

The ideas and results contained in this document are part of my thesis, which will be published as a Stanford computer science technical report in June 1999. Similar ideas applied to the EMD under translation have already been published in the technical report [1]. We begin some intuition, examples, and informal definitions. The Earth Mover's Distance (EMD) is a distance measure between discrete, finite distributions Here x has m=2 masses and y has n=3 masses. The circle centers are the points (mass locations) of the distributions. The area of a circle is proportional to the weight at its center point. The total weight of x is w S =sum i=1..m w i , and the total weight of y is u S =sum j=1..n u j. In the example above, the distributions have equal total weight w S =u S =1. Although the EMD does not require equal-weight distributions, let us begin our discussion with this assumption. The EMD between two equal-weight distributions is proportional to the amount of work needed to morph one distribution into the other. We morph x into y by transporting mass from the x mass locations to the y mass locations until x has been rearranged to look exactly like y. An example morph is shown below. The amount of mass transported from x i to y j is denoted f ij , and is called a flow between x i and y j. The work done to transport an amount of mass f ij from x i to y j is the product of the f ij and the distance d ij =d(x i ,y j) between x i and y j. The total amount of work to morph x into y by the flow F=(f ij) is the sum of the individual works: The flow in the previous example is not an optimal flow between the given distributions. A minimum work flow is shown below. The total amount of work done by this flow is 0.23*155.7 + 0.26*277.0 + 0.25*252.3 + 0.26*198.2 = 222.4. The EMD between equal-weight distributions is the minimum work to morph one into the other, divided by the total weight of the distributions. The normalization by the total weight makes the EMD equal to the average distance travelled by mass during an optimal (i.e. work minimizing) flow. The EMD does not change if all the weights in both distributions …