1,510 research outputs found
Relation Embedding with Dihedral Group in Knowledge Graph
Link prediction is critical for the application of incomplete knowledge graph
(KG) in the downstream tasks. As a family of effective approaches for link
predictions, embedding methods try to learn low-rank representations for both
entities and relations such that the bilinear form defined therein is a
well-behaved scoring function. Despite of their successful performances,
existing bilinear forms overlook the modeling of relation compositions,
resulting in lacks of interpretability for reasoning on KG. To fulfill this
gap, we propose a new model called DihEdral, named after dihedral symmetry
group. This new model learns knowledge graph embeddings that can capture
relation compositions by nature. Furthermore, our approach models the relation
embeddings parametrized by discrete values, thereby decrease the solution space
drastically. Our experiments show that DihEdral is able to capture all desired
properties such as (skew-) symmetry, inversion and (non-) Abelian composition,
and outperforms existing bilinear form based approach and is comparable to or
better than deep learning models such as ConvE.Comment: ACL 201
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio
Musical Actions of Dihedral Groups
The sequence of pitches which form a musical melody can be transposed or
inverted. Since the 1970s, music theorists have modeled musical transposition
and inversion in terms of an action of the dihedral group of order 24. More
recently music theorists have found an intriguing second way that the dihedral
group of order 24 acts on the set of major and minor chords. We illustrate both
geometrically and algebraically how these two actions are {\it dual}. Both
actions and their duality have been used to analyze works of music as diverse
as Hindemith and the Beatles.Comment: 27 pages, 11 figures. To appear in the American Mathematical Monthly
NagE: Non-Abelian Group Embedding for Knowledge Graphs
We demonstrated the existence of a group algebraic structure hidden in
relational knowledge embedding problems, which suggests that a group-based
embedding framework is essential for designing embedding models. Our
theoretical analysis explores merely the intrinsic property of the embedding
problem itself hence is model-independent. Motivated by the theoretical
analysis, we have proposed a group theory-based knowledge graph embedding
framework, in which relations are embedded as group elements, and entities are
represented by vectors in group action spaces. We provide a generic recipe to
construct embedding models associated with two instantiating examples: SO3E and
SU2E, both of which apply a continuous non-Abelian group as the relation
embedding. Empirical experiments using these two exampling models have shown
state-of-the-art results on benchmark datasets.Comment: work accepted the 29th ACM International Conference on Information
and Knowledge Managemen
Oka's conjecture on irreducible plane sextics
We partially prove and partially disprove Oka's conjecture on the fundamental
group/Alexander polynomial of an irreducible plane sextic. Among other results,
we enumerate all irreducible sextics with simple singularities admitting
dihedral coverings and find examples of Alexander equivalent Zariski pairs of
irreducible sextics.Comment: Final version accepted for publicatio
Lower algebraic K-theory of certain reflection groups
For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the
property that all interior angles between incident faces are integral
submultiples of Pi, there is a naturally associated Coxeter group generated by
reflections in the faces. Furthermore, this Coxeter group is a lattice inside
the isometry group of hyperbolic 3-space, with fundamental domain the original
polyhedron P. In this paper, we provide a procedure for computing the lower
algebraic K-theory of the integral group ring of such Coxeter lattices in terms
of the geometry of the polyhedron P. As an ingredient in the computation, we
explicitly calculate some of the lower K-groups of the dihedral groups and the
product of dihedral groups with the cyclic group of order two.Comment: 35 pages, 2 figure
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