Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Bethe and Sinkhorn Permanents of Low Rank Matrices and Implications
  for Profile Maximum Likelihood   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.
  We achieve these results by providing new bounds on the quality of
approximation of the Bethe and Sinkhorn permanents (Vontobel, 2012 and 2014).
We show that each of these are $\exp(O(k \log(N/k)))$ approximations to the
permanent of $N \times N$ matrices with non-negative rank at most $k$,
improving upon the previous known bounds of $\exp(O(N))$. To obtain our results
on PML, we exploit the fact that the PML objective is proportional to the
permanent of a certain Vandermonde matrix with $\sqrt{n}$ distinct columns,
i.e. with non-negative rank at most $\sqrt{n}$. As a by-product of our work we
establish a surprising connection between the convex relaxation in prior work
(CSS19) and the well-studied Bethe and Sinkhorn approximations.

4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex