Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Structure Learning of $H$-colorings   We study the structure learning problem for $H$-colorings, an important class
of Markov random fields that capture key combinatorial structures on graphs,
including proper colorings and independent sets, as well as spin systems from
statistical physics. The learning problem is as follows: for a fixed (and
known) constraint graph $H$ with $q$ colors and an unknown graph $G=(V,E)$ with
$n$ vertices, given uniformly random $H$-colorings of $G$, how many samples are
required to learn the edges of the unknown graph $G$? We give a
characterization of $H$ for which the problem is identifiable for every $G$,
i.e., we can learn $G$ with an infinite number of samples. We also show that
there are identifiable constraint graphs for which one cannot hope to learn
every graph $G$ efficiently.
  We focus particular attention on the case of proper vertex $q$-colorings of
graphs of maximum degree $d$ where intriguing connections to statistical
physics phase transitions appear. We prove that in the tree uniqueness region
(when $q>d$) the problem is identifiable and we can learn $G$ in ${\rm
poly}(d,q) \times O(n^2\log{n})$ time. In contrast for soft-constraint systems,
such as the Ising model, the best possible running time is exponential in $d$.
In the tree non-uniqueness region (when $q\leq d$) we prove that the problem is
not identifiable and thus $G$ cannot be learned. Moreover, when $q<d-\sqrt{d} +
\Theta(1)$ we prove that even learning an equivalent graph (any graph with the
same set of $H$-colorings) is computationally hard---sample complexity is
exponential in $n$ in the worst case. We further explore the connection between
the efficiency/hardness of the structure learning problem and the
uniqueness/non-uniqueness phase transition for general $H$-colorings and prove
that under the well-known Dobrushin uniqueness condition, we can learn $G$ in
${\rm poly}(d,q)\times O(n^2\log{n})$ time.

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The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph