Space-Efficient Approximation Scheme for Circular Earth Mover Distance

Abstract

The Earth Mover Distance (EMD) between point sets A and B is the minimum cost of a bipartite matching between A and B. EMD is an important measure for estimating similarities between objects with quantifiable features and has important applications in several areas including computer vision. The streaming complexity of approximating EMD between point sets in a two-dimensional discretized grid is an important open problem proposed in [8,9].

We study the problem of approximating EMD in the streaming model, when points lie on a discretized circle. Computing the EMD in this setting has applications to computer vision [13] and can be seen as a special case of computing EMD on a discretized grid. We achieve a (1 ±ε) approximation for EMD in $\tilde O(\varepsilon^{-3})$ space, for every 0 < ε < 1. To our knowledge, this is the first streaming algorithm for a natural and widely applied EMD model that matches the space bound asked in [9].

This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, and the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174. The authors acknowledge support from the Danish National Research Foundation and The National Science Foundation of China (under the grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, within which this work was performed. The authors also acknowledge support from the Center for Research in Foundations of Electronic Markets (CFEM), supported by the Danish Strategic Research Council. Joshua Brody is also supported in part by the Danish Council for Independent Research grant LOBO 438146. Xiaoming Sun is also supported in part by the National Natural Science Foundation of China Grant 61170062 and Tsinghua University Initiative Scientific Research Program 2009THZ02120.