Name
Abstract | ||
|---|---|---|
We present a framework for speeding up the time it takes to sample from discrete distributions $\mu$ defined over subsets of size $k$ of a ground set of $n$ elements, in the regime $k\ll n$. We show that having estimates of marginals $\mathbb{P}_{S\sim \mu}[i\in S]$, the task of sampling from $\mu$ can be reduced to sampling from distributions $\nu$ supported on size $k$ subsets of a ground set of only $n^{1-\alpha}\cdot \operatorname{poly}(k)$ elements. Here, $1/\alpha\in [1, k]$ is the parameter of entropic independence for $\mu$. Further, the sparsified distributions $\nu$ are obtained by applying a sparse (mostly $0$) external field to $\mu$, 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 $\mu$, and in return reduce the amortized cost needed to produce many samples from the distribution $\mu$, as is often needed in upstream tasks such as counting and inference. For a wide range of distributions where $\alpha=\Omega(1)$, our result reduces the domain size, and as a corollary, the cost-per-sample, by a $\operatorname{poly}(n)$ 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 $\alpha=1$). 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. |
Year | DOI | Venue |
|---|---|---|
2022 | 10.4230/LIPIcs.ITCS.2022.5 | ITCS |
DocType | Citations | PageRank |
Conference | 0 | 0.34 |
References | Authors | |
0 | 2 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Nima Anari | 1 | 0 | 0.68 |
| Michał Dereziński | 2 | 0 | 0.68 |