A new reduction from high-dimensional EMD to Closest Pair yields a subquadratic (1+ε)-approximation algorithm with running time n^{2-Ω~(ε^{1/3})}, but the proof contains a likely gap in the randomized rounding argument.
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Approximating High-Dimensional Earth Mover's Distance as Fast as Closest Pair
A new reduction from high-dimensional EMD to Closest Pair yields a subquadratic (1+ε)-approximation algorithm with running time n^{2-Ω~(ε^{1/3})}, but the proof contains a likely gap in the randomized rounding argument.