Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Entropic Independence II: Optimal Sampling and Concentration via
  Restricted Modified Log-Sobolev Inequalities   We introduce a framework for obtaining tight mixing times for Markov chains
based on what we call restricted modified log-Sobolev inequalities. Modified
log-Sobolev inequalities (MLSI) quantify the rate of relative entropy
contraction for the Markov operator, and are notoriously difficult to
establish. However, infinitesimally close to stationarity, entropy contraction
becomes equivalent to variance contraction, a.k.a. a Poincare inequality, which
is significantly easier to establish through, e.g., spectral analysis.
Motivated by this observation, we study restricted modified log-Sobolev
inequalities that guarantee entropy contraction not for all starting
distributions, but for those in a large neighborhood of the stationary
distribution. We show how to sample from the hardcore and Ising models on
$n$-node graphs that have a constant $\delta$ relative gap to the
tree-uniqueness threshold, in nearly-linear time $\widetilde O_{\delta}(n)$.
Notably, our bound does not depend on the maximum degree $\Delta$, and is
therefore optimal even for high-degree graphs. This improves on prior mixing
time bounds of $\widetilde O_{\delta, \Delta}(n)$ and $\widetilde
O_{\delta}(n^2)$, established via (non-restricted) modified log-Sobolev and
Poincare inequalities respectively. We further show that optimal concentration
inequalities can still be achieved from the restricted form of modified
log-Sobolev inequalities. To establish restricted entropy contraction, we
extend the entropic independence framework of Anari, Jain, Koehler, Pham, and
Vuong to the setting of distributions that are spectrally independent under a
restricted set of external fields. We also develop an orthogonal trick that
might be of independent interest: utilizing Bernoulli factories we show how to
implement Glauber dynamics updates on high-degree graphs in $O(1)$ time,
assuming standard adjacency array representation of the graph.

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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: stat