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 <tex>$\mu$</tex> is a distribution over subsets <tex>$\mathcal{S}$</tex> of a ground set [<tex>$n$</tex>], we say that <tex>$\mu$</tex> is in isotropic position if <tex>$\mathbb{P}_{\mathcal{S}\sim\mu}[e\in \mathcal{S}]$</tex> is the same for all <tex>$e\in[n]$</tex>. We design a new approximate sampling algorithm that leverages isotropy for the class of distributions <tex>$\mu$</tex> 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 <tex>$\mu$</tex> is in approximately isotropic position, the running time of our algorithm depends polynomially on the size of the set <tex>$\mathcal{S}$</tex>, and only logarithmically on <tex>$n$</tex>. When <tex>$n$</tex> is much larger than the size of <tex>$\mathcal{S}$</tex>, this is significantly faster than prior algorithms, and can even be sublinear in <tex>$n$</tex>. We then show how to transform a non-isotropic <tex>$\mu$</tex> into an equivalent approximately isotropic form with a polynomial-time pre-processing 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 <tex>$k$</tex> over a ground set of <tex>$n$</tex> elements to within a factor of <tex>$1+\epsilon$</tex> in time <tex>$O((n+1/\epsilon^{2}) \cdot \text{poly}(k,\log n))$</tex>. This is the first algorithm that runs in nearly linear time for fixed rank <tex>$k$</tex>, and achieves an inverse polynomially low approximation error. The full version of this paper is available at: https://arxiv.org/abs/2004.09079