25,866 research outputs found
Blowup Equations for Refined Topological Strings
G\"{o}ttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the
Nekrasov partition function of five dimensional supersymmetric
gauge theories compactified on a circle, which via geometric engineering
correspond to the refined topological string theory on geometries. In
this paper, we study the K-theoretic blowup equations for general local
Calabi-Yau threefolds. We find that both vanishing and unity blowup equations
exist for the partition function of refined topological string, and the crucial
ingredients are the fields introduced in our previous paper. These
blowup equations are in fact the functional equations for the partition
function and each of them results in infinite identities among the refined free
energies. Evidences show that they can be used to determine the full refined
BPS invariants of local Calabi-Yau threefolds. This serves an independent and
sometimes more powerful way to compute the partition function other than the
refined topological vertex in the A-model and the refined holomorphic anomaly
equations in the B-model. We study the modular properties of the blowup
equations and provide a procedure to determine all the vanishing and unity fields from the polynomial part of refined topological string at large
radius point. We also find that certain form of blowup equations exist at
generic loci of the moduli space.Comment: 85 pages. v2: Journal versio
Continual Local Training for Better Initialization of Federated Models
Federated learning (FL) refers to the learning paradigm that trains machine
learning models directly in the decentralized systems consisting of smart edge
devices without transmitting the raw data, which avoids the heavy communication
costs and privacy concerns. Given the typical heterogeneous data distributions
in such situations, the popular FL algorithm \emph{Federated Averaging}
(FedAvg) suffers from weight divergence and thus cannot achieve a competitive
performance for the global model (denoted as the \emph{initial performance} in
FL) compared to centralized methods. In this paper, we propose the local
continual training strategy to address this problem. Importance weights are
evaluated on a small proxy dataset on the central server and then used to
constrain the local training. With this additional term, we alleviate the
weight divergence and continually integrate the knowledge on different local
clients into the global model, which ensures a better generalization ability.
Experiments on various FL settings demonstrate that our method significantly
improves the initial performance of federated models with few extra
communication costs.Comment: This paper has been accepted to 2020 IEEE International Conference on
Image Processing (ICIP 2020
Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent
We continue our study of the inventory accumulation introduced by Sheffield
(2011), which encodes a random planar map decorated by a collection of loops
sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various
\emph{local estimates} for the inventory accumulation model, i.e., estimates
for the precise number of symbols of a given type in a reduced word sampled
from the model. Using our estimates, we obtain the scaling limit of the
associated two-dimensional random walk conditioned on the event that it stays
in the first quadrant for one unit of time and ends up at a particular position
in the interior of the first quadrant. We also obtain the exponent for the
probability that a word of length sampled from the inventory accumulation
model corresponds to an empty reduced word, which is equivalent to an
asymptotic formula for the partition function of the critical FK planar map
model. The estimates of this paper will be used in a subsequent paper to obtain
the scaling limit of the lattice walk associated with a finite-volume FK planar
map.Comment: 49 pages, 2 figures; final version published in EJP. Changes include
significantly approved exposition and relation to partition functio
Ergodicity of the Airy line ensemble
In this paper, we establish the ergodicity of the Airy line ensemble. This
shows that it is the only candidate for Conjecture 3.2 in [3], regarding the
classification of ergodic line ensembles satisfying a certain Brownian Gibbs
property after a parabolic shift.Comment: argument for Proposition 1.13 is revised, the structure of the
introduction is rearrange
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