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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 where $$k$$ is much smaller than $$n$$. We show that if one has access to estimates of marginals $$\mathbb{P}_{S\sim \mu}[i\in S]$$, then the task of sampling from $$\mu$$ can be reduced to sampling from related 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, our algorithm only requires sparsified distributions $$\nu$$ that are obtained by applying a sparse (mostly $$0$$) external field to $$\mu$$, an operation that for many distributions $$\mu$$ of interest, 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.