392 research outputs found
The convexity package for Hamiltonian actions on conformal symplectic manifolds
Consider a Hamiltonian action of a compact connected Lie group on a conformal
symplectic manifold. We prove a convexity theorem for the moment map under the
assumption that the action is of Lee type, which establishes an analog of
Kirwan's convexity theorem in conformal symplectic geometry.Comment: 31 pages, 1 figure. Appendix on conformal presymplectic manifolds
added. Minor mistakes correcte
Resource-Adaptive Newton's Method for Distributed Learning
Distributed stochastic optimization methods based on Newton's method offer
significant advantages over first-order methods by leveraging curvature
information for improved performance. However, the practical applicability of
Newton's method is hindered in large-scale and heterogeneous learning
environments due to challenges such as high computation and communication costs
associated with the Hessian matrix, sub-model diversity, staleness in training,
and data heterogeneity. To address these challenges, this paper introduces a
novel and efficient algorithm called RANL, which overcomes the limitations of
Newton's method by employing a simple Hessian initialization and adaptive
assignments of training regions. The algorithm demonstrates impressive
convergence properties, which are rigorously analyzed under standard
assumptions in stochastic optimization. The theoretical analysis establishes
that RANL achieves a linear convergence rate while effectively adapting to
available resources and maintaining high efficiency. Unlike traditional
first-order methods, RANL exhibits remarkable independence from the condition
number of the problem and eliminates the need for complex parameter tuning.
These advantages make RANL a promising approach for distributed stochastic
optimization in practical scenarios
And\^o dilations for a pair of commuting contractions: two explicit constructions and functional models
One of the most important results in operator theory is And\^o's \cite{ando}
generalization of dilation theory for a single contraction to a pair of
commuting contractions acting on a Hilbert space. While there are two explicit
constructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the
minimal isometric dilation of a single contraction, there was no such explicit
construction of an And\^o dilation for a commuting pair of
contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In
this paper, we give two new proofs of And\^o's dilation theorem by giving both
Sch\"affer-type and Douglas-type explicit constructions of an And\^o dilation
with function-theoretic interpretation, for the general case. The results, in
particular, give a complete description of all possible factorizations of a
given contraction into the product of two commuting contractions. Unlike
the one-variable case, two minimal And\^o dilations need not be unitarily
equivalent. However, we show that the compressions of the two And\^o dilations
constructed in this paper to the minimal dilation spaces of the contraction
, are unitarily equivalent.
In the special case when the product is pure, i.e., if strongly, an And\^o dilation was constructed recently in \cite{D-S-S},
which, as this paper will show, is a corollary to the Douglas-type
construction.
We define a notion of characteristic triple for a pair of commuting
contractions and a notion of coincidence for such triples. We prove that two
pairs of commuting contractions with their products being pure contractions are
unitarily equivalent if and only if their characteristic triples coincide. We
also characterize triples which qualify as the characteristic triple for some
pair of commuting contractions such that is a pure
contraction.Comment: 24 page
Contextual Dictionary Lookup for Knowledge Graph Completion
Knowledge graph completion (KGC) aims to solve the incompleteness of
knowledge graphs (KGs) by predicting missing links from known triples, numbers
of knowledge graph embedding (KGE) models have been proposed to perform KGC by
learning embeddings. Nevertheless, most existing embedding models map each
relation into a unique vector, overlooking the specific fine-grained semantics
of them under different entities. Additionally, the few available fine-grained
semantic models rely on clustering algorithms, resulting in limited performance
and applicability due to the cumbersome two-stage training process. In this
paper, we present a novel method utilizing contextual dictionary lookup,
enabling conventional embedding models to learn fine-grained semantics of
relations in an end-to-end manner. More specifically, we represent each
relation using a dictionary that contains multiple latent semantics. The
composition of a given entity and the dictionary's central semantics serves as
the context for generating a lookup, thus determining the fine-grained
semantics of the relation adaptively. The proposed loss function optimizes both
the central and fine-grained semantics simultaneously to ensure their semantic
consistency. Besides, we introduce two metrics to assess the validity and
accuracy of the dictionary lookup operation. We extend several KGE models with
the method, resulting in substantial performance improvements on widely-used
benchmark datasets
Thermal Hall effect in insulating quantum materials
The emerging field of quantum materials involves an exciting new class of
materials in which charge, spin, orbital, and lattice degrees of freedom are inter-
twined, exhibiting a plethora of exotic physical properties. Quantum materials
include, but are not limited to, superconductors, topological quantum matter,
and systems with frustrated spins, which enable a wide range of potential
applications in biomedicine, energy transport and conversion, quantum sensing,
and quantum information processing.S.G. and X.C. acknowledge the support from National Science Foundation under grant No.
2144328.
J.Z. acknowledges the support from National Science Foundation through the Center
for Dynamics and Control of Materials: an NSF MRSEC unnder Cooperative Agreement
No. DMR-1720595.Center for Dynamics and Control of Material
- …