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Suppose we are given observations, where each observation is drawnindependently from one of $k$ known distributions. The goal is to match eachobservation to the distribution from which it was drawn. We observe that themaximum likelihood estimator (MLE) for this problem can be computed usingweighted bipartite matching, even when $n$, the number of observations perdistribution, exceeds one. This is achieved by instantiating $n$ duplicates ofeach distribution node. However, in the regime where the number of observationsper distribution is much larger than the number of distributions, the Hungarianmatching algorithm for computing the weighted bipartite matching requires$\mathcal O(n^3)$ time. We introduce a novel randomized matching algorithm thatreduces the runtime to $\tilde{\mathcal O}(n^2)$ by sparsifying the originalgraph, returning the exact MLE with high probability. Next, we give statisticaljustification for using the MLE by bounding the excess risk of the MLE, wherethe loss is defined as the negative log-likelihood. We test these bounds forthe case of isotropic Gaussians with equal covariances and whose means areseparated by a distance $\eta$, and find (1) that $\gg \log k$ separationsuffices to drive the proportion of mismatches of the MLE to 0, and (2) thatthe expected fraction of mismatched observations goes to zero at rate $\mathcalO({(\log k)}^2/\eta^2)$.

Matching Observations to Distributions: Efficient Estimation via Sparsified Hungarian Algorithm