26,650 research outputs found
Quantum Electrodynamics at Large Distances II: Nature of the Dominant Singularities
Accurate calculations of macroscopic and mesoscopic properties in quantum
electrodynamics require careful treatment of infrared divergences: standard
treatments introduce spurious large-distances effects. A method for computing
these properties was developed in a companion paper. That method depends upon a
result obtained here about the nature of the singularities that produce the
dominant large-distance behaviour. If all particles in a quantum field theory
have non-zero mass then the Landau-Nakanishi diagrams give strong conditions on
the singularities of the scattering functions. These conditions are severely
weakened in quantum electrodynamics by effects of points where photon momenta
vanish. A new kind of Landau-Nakanishi diagram is developed here. It is geared
specifically to the pole-decomposition functions that dominate the macroscopic
behaviour in quantum electrodynamics, and leads to strong results for these
functions at points where photon momenta vanish.Comment: 40 pages, 11 encapsulated postscript figures, latexed,
math_macros.tex can be found on Archive. full postscript available from
http://theorl.lbl.gov/www/theorgroup/papers/35972.p
Graph Convolutional Neural Networks for Web-Scale Recommender Systems
Recent advancements in deep neural networks for graph-structured data have
led to state-of-the-art performance on recommender system benchmarks. However,
making these methods practical and scalable to web-scale recommendation tasks
with billions of items and hundreds of millions of users remains a challenge.
Here we describe a large-scale deep recommendation engine that we developed and
deployed at Pinterest. We develop a data-efficient Graph Convolutional Network
(GCN) algorithm PinSage, which combines efficient random walks and graph
convolutions to generate embeddings of nodes (i.e., items) that incorporate
both graph structure as well as node feature information. Compared to prior GCN
approaches, we develop a novel method based on highly efficient random walks to
structure the convolutions and design a novel training strategy that relies on
harder-and-harder training examples to improve robustness and convergence of
the model. We also develop an efficient MapReduce model inference algorithm to
generate embeddings using a trained model. We deploy PinSage at Pinterest and
train it on 7.5 billion examples on a graph with 3 billion nodes representing
pins and boards, and 18 billion edges. According to offline metrics, user
studies and A/B tests, PinSage generates higher-quality recommendations than
comparable deep learning and graph-based alternatives. To our knowledge, this
is the largest application of deep graph embeddings to date and paves the way
for a new generation of web-scale recommender systems based on graph
convolutional architectures.Comment: KDD 201
Statistical Mechanics of maximal independent sets
The graph theoretic concept of maximal independent set arises in several
practical problems in computer science as well as in game theory. A maximal
independent set is defined by the set of occupied nodes that satisfy some
packing and covering constraints. It is known that finding minimum and
maximum-density maximal independent sets are hard optimization problems. In
this paper, we use cavity method of statistical physics and Monte Carlo
simulations to study the corresponding constraint satisfaction problem on
random graphs. We obtain the entropy of maximal independent sets within the
replica symmetric and one-step replica symmetry breaking frameworks, shedding
light on the metric structure of the landscape of solutions and suggesting a
class of possible algorithms. This is of particular relevance for the
application to the study of strategic interactions in social and economic
networks, where maximal independent sets correspond to pure Nash equilibria of
a graphical game of public goods allocation
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