78,109 research outputs found
On a type of semi-sub-Riemannian connection on a sub-Riemannian manifold
The authors first in this paper define a semi-symmetric metric non-holonomic
connection (called in briefly a semi-sub-Riemannian connection) on
sub-Riemannian manifolds, and study the relations between sub-Riemannian
connections and semi-sub-Riemannian connections. An invariant under a
connection transformation is obtained. The authors then
further deduce a sufficient and necessary condition that a sub-Riemannian
manifold associated with a semi-sub-Riemannian connection is flat, and derive
that a sub-Riemannian manifold with vanishing curvature with respect to
semi-sub-Riemannnian connection is a group manifold if and only if it is of
constant curvature.Comment: 18 page
On SNS-Riemannian connections in sub-Riemannian manifolds
The authors define a SNS (semi-nearly-sub)-Riemannian connection on nearly
sub-Riemannian manifolds and study the geometric properties of such a
connection, and obtain the natures of horizontal curvature tensors between
horizontal sub-Riemannian connection and SNS-Riemannian connection. The authors
further investigate the geometric characteristics of the projective
SNS-Riemannian connection, and obtain a necessary and sufficient condition for
a nearly sub-Riemannian manifold being projectively flat.Comment: It has not been published by no
Minimum codegree threshold for Hamilton l-cycles in k-uniform hypergraphs
For , we show that for sufficiently large , every
-uniform hypergraph on vertices with minimum codegree at least contains a Hamilton -cycle. This codegree condition is
best possible and improves on work of H\`an and Schacht who proved an
asymptotic result.Comment: 22 pages, 0 figure. Accepted for publication in JCTA. arXiv admin
note: text overlap with arXiv:1307.369
A Sober Look at Spectral Learning
Spectral learning recently generated lots of excitement in machine learning,
largely because it is the first known method to produce consistent estimates
(under suitable conditions) for several latent variable models. In contrast,
maximum likelihood estimates may get trapped in local optima due to the
non-convex nature of the likelihood function of latent variable models. In this
paper, we do an empirical evaluation of spectral learning (SL) and expectation
maximization (EM), which reveals an important gap between the theory and the
practice. First, SL often leads to negative probabilities. Second, EM often
yields better estimates than spectral learning and it does not seem to get
stuck in local optima. We discuss how the rank of the model parameters and the
amount of training data can yield negative probabilities. We also question the
common belief that maximum likelihood estimators are necessarily inconsistent
Minimum vertex degree threshold for -tiling
We prove that the vertex degree threshold for tiling \C_4^3 (the 3-uniform
hypergraph with four vertices and two triples) in a 3-uniform hypergraph on
vertices is ,
where if and otherwise. This result is
best possible, and is one of the first results on vertex degree conditions for
hypergraph tiling.Comment: 16 pages, 0 figure. arXiv admin note: text overlap with
arXiv:0903.2867 by other author
Frank-Wolfe Optimization for Symmetric-NMF under Simplicial Constraint
Symmetric nonnegative matrix factorization has found abundant applications in
various domains by providing a symmetric low-rank decomposition of nonnegative
matrices. In this paper we propose a Frank-Wolfe (FW) solver to optimize the
symmetric nonnegative matrix factorization problem under a simplicial
constraint, which has recently been proposed for probabilistic clustering.
Compared with existing solutions, this algorithm is simple to implement, and
has no hyperparameters to be tuned. Building on the recent advances of FW
algorithms in nonconvex optimization, we prove an
convergence rate to -approximate KKT points, via a tight bound
on the curvature constant, which matches the best known result in
unconstrained nonconvex setting using gradient methods. Numerical results
demonstrate the effectiveness of our algorithm. As a side contribution, we
construct a simple nonsmooth convex problem where the FW algorithm fails to
converge to the optimum. This result raises an interesting question about
necessary conditions of the success of the FW algorithm on convex problems.Comment: In Proceedings of the Thirty-Fourth Conference on Uncertainty in
Artificial Intelligence, 201
Forbidding Hamilton cycles in uniform hypergraphs
For , we give a new lower bound for the minimum -degree
threshold that guarantees a Hamilton -cycle in -uniform hypergraphs.
When and , this bound is larger than the conjectured
minimum -degree threshold for perfect matchings and thus disproves a
well-known conjecture of R\"odl and Ruci\'nski. Our (simple) construction
generalizes a construction of Katona and Kierstead and the space barrier for
Hamilton cycles.Comment: 6 pages, 0 figur
Superspecies and their representations
Superspecies are introduced to provide the nice constructions of all
finite-dimensional superalgebras. All acyclic superspecies, or equivalently all
finite-dimensional (gr-basic) gr-hereditary superalgebras, are classified
according to their graded representation types. To this end, graded
equivalence, graded representation type and graded species are introduced for
finite group graded algebras.Comment: 36 page
On multipartite Hajnal-Szemer\'edi theorems
Let be a -partite graph with vertices in parts such that each
vertex is adjacent to at least vertices in each of the other
parts. Magyar and Martin \cite{MaMa} proved that for , if and is sufficiently large, then contains a -factor (a
spanning subgraph consisting of vertex-disjoint copies of ) except
that is one particular graph. Martin and Szemer\'edi \cite{MaSz} proved
that contains a -factor when and is
sufficiently large. Both results were proved by the Regularity Lemma. In this
paper we give a proof of these two results by the absorbing method. Our
absorbing lemma actually works for all .Comment: 15 pages, no figur
Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs
We show that for sufficiently large , every 3-uniform hypergraph on
vertices with minimum vertex degree at least , where if and
if , contains a loose Hamilton
cycle. This degree condition is best possible and improves on the work of
Bu\ss, H\`an and Schacht who proved the corresponding asymptotical result.Comment: 23 pages, 1 figure, Accepted for publication in JCT
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