8,355 research outputs found
The Brownian limit of separable permutations
We study random uniform permutations in an important class of
pattern-avoiding permutations: the separable permutations. We describe the
asymptotics of the number of occurrences of any fixed given pattern in such a
random permutation in terms of the Brownian excursion. In the recent
terminology of permutons, our work can be interpreted as the convergence of
uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion
The Hopf algebra of diagonal rectangulations
We define and study a combinatorial Hopf algebra dRec with basis elements
indexed by diagonal rectangulations of a square. This Hopf algebra provides an
intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter
permutations, which previously had only been described extrinsically as a sub
Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We
describe the natural lattice structure on diagonal rectangulations, analogous
to the Tamari lattice on triangulations, and observe that diagonal
rectangulations index the vertices of a polytope analogous to the
associahedron. We give an explicit bijection between twisted Baxter
permutations and the better-known Baxter permutations, and describe the
resulting Hopf algebra structure on Baxter permutations.Comment: Very minor changes from version 1, in response to comments by
referees. This is the final version, to appear in JCTA. 43 pages, 17 figure
Identifiability and Unmixing of Latent Parse Trees
This paper explores unsupervised learning of parsing models along two
directions. First, which models are identifiable from infinite data? We use a
general technique for numerically checking identifiability based on the rank of
a Jacobian matrix, and apply it to several standard constituency and dependency
parsing models. Second, for identifiable models, how do we estimate the
parameters efficiently? EM suffers from local optima, while recent work using
spectral methods cannot be directly applied since the topology of the parse
tree varies across sentences. We develop a strategy, unmixing, which deals with
this additional complexity for restricted classes of parsing models
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