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

Abstract

The Earth Mover’s Distance (EMD) on weighted point sets is a distance measure with many applications. Since there are no known exact algorithms to compute the minimum EMD under transformations, it is useful to estimate the minimum EMD under various classes of transformations. For weighted point sets in the plane, we will show a 2-approximation algorithm for translations, a 4-approximation algorithm for rigid motions and an 8-approximation algorithm for similarity transformations. The runtime for translations is O(T EMD(n,m)), the runtime of the latter two algorithms is O(nmT EMD(n,m)), where T EMD(n,m) is the time to compute the EMD between two fixed weighted point sets with n and m points, respectively. All these algorithms are based on a more general structure, namely on reference points. This leads to elegant generalizations to higher dimensions. We give a comprehensive discussion of reference points for weighted point sets with respect to the EMD.KeywordsReference PointApproximation AlgorithmDistance MeasureSimilarity TransformationArbitrary DimensionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.