Isotropy and Log-Concave Polynomials: Accelerated Sampling and High-Precision Counting of Matroid Bases

We define a notion of isotropy for discrete set distributions. If $\mu$ is a distribution over subsets $S$ of a ground set $[n]$, we say that $\mu$ is in isotropic position if $P[e \in S]$ is the same for all $e\in [n]$. We design a new approximate sampling algorithm that leverages isotropy for the class of distributions $\mu$ that have a log-concave generating polynomial; this class includes determinantal point processes, strongly Rayleigh distributions, and uniform distributions over matroid bases. We show that when $\mu$ is in approximately isotropic position, the running time of our algorithm depends polynomially on the size of the set $S$, and only logarithmically on $n$. When $n$ is much larger than the size of $S$, this is significantly faster than prior algorithms, and can even be sublinear in $n$. We then show how to transform a non-isotropic $\mu$ into an equivalent approximately isotropic form with a polynomial-time preprocessing step, accelerating subsequent sampling times. The main new ingredient enabling our algorithms is a class of negative dependence inequalities that may be of independent interest. As an application of our results, we show how to approximately count bases of a matroid of rank $k$ over a ground set of $n$ elements to within a factor of $1+\epsilon$ in time $O((n+1/\epsilon^2)\cdot poly(k, \log n))$. This is the first algorithm that runs in nearly linear time for fixed rank $k$, and achieves an inverse polynomially low approximation error.