# Domain Sparsification of Discrete Distributions using Entropic Independence

arXiv: https://arxiv.org/abs/2109.06442 · HTML: https://mlsanity.com/paper/2109.06442/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2021.

## Built on by (most cited first, 2 of 2)

- [Recent and Upcoming Developments in Randomized Numerical Linear Algebra for Machine Learning](https://mlsanity.com/paper/2406.11151/) (2024)
- [Optimal Sublinear Sampling of Spanning Trees and Determinantal Point Processes via Average-Case Entropic Independence](https://mlsanity.com/paper/2204.02570/) (2022)

## Abstract (arXiv)

We present a framework for speeding up the time it takes to sample from discrete distributions μ defined over subsets of size k of a ground set of n elements, in the regime kll n. We show that having estimates of marginals mathbbPS~ μ[i∈ S], the task of sampling from μ can be reduced to sampling from distributions nu supported on size k subsets of a ground set of only n1-α· operatornamepoly(k) elements. Here, 1/α∈ [1, k] is the parameter of entropic independence for μ. Further, the sparsified distributions nu are obtained by applying a sparse (mostly 0) external field to μ, an operation that often retains algorithmic tractability of sampling from nu. This phenomenon, which we dub domain sparsification, allows us to pay a one-time cost of estimating the marginals of μ, and in return reduce the amortized cost needed to produce many samples from the distribution μ, as is often needed in upstream tasks such as counting and inference. For a wide range of distributions where α=Omega(1), our result reduces the domain size, and as a corollary, the cost-per-sample, by a operatornamepoly(n) factor. Examples include monomers in a monomer-dimer system, non-symmetric determinantal point processes, and partition-constrained Strongly Rayleigh measures. Our work significantly extends the reach of prior work of Anari and Derezi'nski who obtained domain sparsification for distributions with a log-concave generating polynomial (corresponding to α=1). As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show that constant-factor approximation is enough for domain sparsification, improving over O(1/k) relative error established in prior work.