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# The Bethe and Sinkhorn Permanents of Low Rank Matrices and Implications for Profile Maximum Likelihood

*Proceedings of Thirty Fourth Conference on Learning Theory*, PMLR 134:93-158, 2021.

#### Abstract

In this paper we consider the problem of computing the likelihood of the profile of a discrete distribution, i.e., the probability of observing the multiset of element frequencies, and computing a profile maximum likelihood (PML) distribution, i.e., a distribution with the maximum profile likelihood. For each problem we provide polynomial time algorithms that given $n$ i.i.d. samples from a discrete distribution, achieve an approximation factor of $\exp\left(-O(\sqrt{n} \log n) \right)$, improving upon the previous best-known bound achievable in polynomial time of $\exp(-O(n^{2/3} \log n))$ (Charikar, Shiragur and Sidford, 2019). Through the work of Acharya, Das, Orlitsky and Suresh (2016), this implies a polynomial time universal estimator for symmetric properties of discrete distributions in a broader range of error parameter. To obtain our results on PML we establish new connections between PML and the well-studied Bethe and Sinkhorn approximations to the permanent (Vontobel, 2012 and 2014). It is known that the PML objective is proportional to the permanent of a certain Vandermonde matrix (Vontobel, 2012) with $\sqrt{n}$ distinct columns, i.e. with non-negative rank at most $\sqrt{n}$. This allows us to show that the convex approximation to computing PML distributions studied in (Charikar, Shiragur and Sidford, 2019) is governed, in part, by the quality of Sinkhorn approximations to the permanent. We show that both Bethe and Sinkhorn permanents are $\exp(O(k \log(N/k)))$ approximations to the permanent of $N \times N$ matrices with non-negative rank at most $k$. This improves upon the previous known bounds of $\exp(O(N))$ and combining these insights with careful rounding of the convex relaxation yields our results.