Efficient algorithm or approximate algorithm for matching between two sets of vectors in o($n^2$)

Question

I have two sets of vectors of same size $S_1 = \{x_i\},S_2 = \{x_j\}$ and $\forall i\ x_i\in R^d,\forall j \ x_j\in R^d$. I need to find the an assignment between $S_1, S_2$ that will minimize the sum of the $L_1$ distances between the vectors.I know that in case $d=1$ we can sort the sets and then make a greedy match with $O(nlogn)$ complexity. Are there any algorithms (including randomized) that find this match for $d>1$ with $o(n^2)$ complexity?