1,051 research outputs found
Learning Edge Representations via Low-Rank Asymmetric Projections
We propose a new method for embedding graphs while preserving directed edge
information. Learning such continuous-space vector representations (or
embeddings) of nodes in a graph is an important first step for using network
information (from social networks, user-item graphs, knowledge bases, etc.) in
many machine learning tasks.
Unlike previous work, we (1) explicitly model an edge as a function of node
embeddings, and we (2) propose a novel objective, the "graph likelihood", which
contrasts information from sampled random walks with non-existent edges.
Individually, both of these contributions improve the learned representations,
especially when there are memory constraints on the total size of the
embeddings. When combined, our contributions enable us to significantly improve
the state-of-the-art by learning more concise representations that better
preserve the graph structure.
We evaluate our method on a variety of link-prediction task including social
networks, collaboration networks, and protein interactions, showing that our
proposed method learn representations with error reductions of up to 76% and
55%, on directed and undirected graphs. In addition, we show that the
representations learned by our method are quite space efficient, producing
embeddings which have higher structure-preserving accuracy but are 10 times
smaller
Markov convexity and nonembeddability of the Heisenberg group
We compute the Markov convexity invariant of the continuous infinite
dimensional Heisenberg group to show that it is Markov
4-convex and cannot be Markov -convex for any . As Markov convexity
is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a
different proof of the classical theorem of Pansu and Semmes that the
Heisenberg group does not admit a biLipschitz embedding into any Euclidean
space.
The Markov convexity lower bound will follow from exhibiting an explicit
embedding of Laakso graphs into that has distortion
at most . We use this to show that if is a Markov
-convex metric space, then balls of the discrete Heisenberg group
of radius embed into with distortion at least
some constant multiple of
Finally, we show that Markov 4-convexity does not give the optimal distortion
for embeddings of binary trees into by showing that
the distortion is on the order of .Comment: version to appear in Ann. Inst. Fourie
Asymptotically rigid mapping class groups and Thompson's groups
We consider Thompson's groups from the perspective of mapping class groups of
surfaces of infinite type. This point of view leads us to the braided Thompson
groups, which are extensions of Thompson's groups by infinite (spherical) braid
groups. We will outline the main features of these groups and some applications
to the quantization of Teichm\"uller spaces. The chapter provides an
introduction to the subject with an emphasis on some of the authors results.Comment: survey 77
Higher Segal spaces I
This is the first paper in a series on new higher categorical structures
called higher Segal spaces. For every d > 0, we introduce the notion of a
d-Segal space which is a simplicial space satisfying locality conditions
related to triangulations of cyclic polytopes of dimension d. In the case d=1,
we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal
spaces. The starting point of the theory is the observation that Hall algebras,
as previously studied, are only the shadow of a much richer structure governed
by a system of higher coherences captured in the datum of a 2-Segal space. This
2-Segal space is given by Waldhausen's S-construction, a simplicial space
familiar in algebraic K-theory. Other examples of 2-Segal spaces arise
naturally in classical topics such as Hecke algebras, cyclic bar constructions,
configuration spaces of flags, solutions of the pentagon equation, and mapping
class groups.Comment: 221 page
Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness
In this paper, we study algorithmic problems for automaton semigroups and
automaton groups related to freeness and finiteness. In the course of this
study, we also exhibit some connections between the algebraic structure of
automaton (semi)groups and their dynamics on the boundary. First, we show that
it is undecidable to check whether the group generated by a given invertible
automaton has a positive relation, i.e. a relation p = 1 such that p only
contains positive generators. Besides its obvious relation to the freeness of
the group, the absence of positive relations has previously been studied and is
connected to the triviality of some stabilizers of the boundary. We show that
the emptiness of the set of positive relations is equivalent to the dynamical
property that all (directed positive) orbital graphs centered at non-singular
points are acyclic.
Gillibert showed that the finiteness problem for automaton semigroups is
undecidable. In the second part of the paper, we show that this undecidability
result also holds if the input is restricted to be bi-reversible and invertible
(but, in general, not complete). As an immediate consequence, we obtain that
the finiteness problem for automaton subsemigroups of semigroups generated by
invertible, yet partial automata, so called automaton-inverse semigroups, is
also undecidable.
Erratum: Contrary to a statement in a previous version of the paper, our
approach does not show that that the freeness problem for automaton semigroups
is undecidable. We discuss this in an erratum at the end of the paper
All the shapes of spaces: a census of small 3-manifolds
In this work we present a complete (no misses, no duplicates) census for
closed, connected, orientable and prime 3-manifolds induced by plane graphs
with a bipartition of its edge set (blinks) up to edges. Blinks form a
universal encoding for such manifolds. In fact, each such a manifold is a
subtle class of blinks, \cite{lins2013B}. Blinks are in 1-1 correpondence with
{\em blackboard framed links}, \cite {kauffman1991knots, kauffman1994tlr} We
hope that this census becomes as useful for the study of concrete examples of
3-manifolds as the tables of knots are in the study of knots and links.Comment: 31 pages, 17 figures, 38 references. In this version we introduce
some new material concerning composite manifold
Cactus group and monodromy of Bethe vectors
Cactus group is the fundamental group of the real locus of the
Deligne-Mumford moduli space of stable rational curves. This group appears
naturally as an analog of the braid group in coboundary monoidal categories. We
define an action of the cactus group on the set of Bethe vectors of the Gaudin
magnet chain corresponding to arbitrary semisimple Lie algebra .
Cactus group appears in our construction as a subgroup in the Galois group of
Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that
this action is isomorphic to the action of the cactus group on the tensor
product of crystals coming from the general coboundary category formalism. We
prove this conjecture in the case (in
fact, for this case the conjecture almost immediately follows from the results
of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We
also present some conjectures generalizing this result to Bethe vectors of
shift of argument subalgebras and relating the cactus group with the
Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand-Tsetlin
polytope.Comment: 23 pages, sections 2 and 3 revise
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