24,901 research outputs found
On Consistency of Graph-based Semi-supervised Learning
Graph-based semi-supervised learning is one of the most popular methods in
machine learning. Some of its theoretical properties such as bounds for the
generalization error and the convergence of the graph Laplacian regularizer
have been studied in computer science and statistics literatures. However, a
fundamental statistical property, the consistency of the estimator from this
method has not been proved. In this article, we study the consistency problem
under a non-parametric framework. We prove the consistency of graph-based
learning in the case that the estimated scores are enforced to be equal to the
observed responses for the labeled data. The sample sizes of both labeled and
unlabeled data are allowed to grow in this result. When the estimated scores
are not required to be equal to the observed responses, a tuning parameter is
used to balance the loss function and the graph Laplacian regularizer. We give
a counterexample demonstrating that the estimator for this case can be
inconsistent. The theoretical findings are supported by numerical studies.Comment: This paper is accepted by 2019 IEEE 39th International Conference on
Distributed Computing Systems (ICDCS
Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
For a physical field theory, the tree-level amplitudes should possess only
single poles. However, when computing amplitudes with Cachazo-He-Yuan (CHY)
formulation, individual terms in the intermediate steps will contribute
higher-order poles. In this paper, we investigate the cancelation of
higher-order poles in CHY formula with Pfaffian as the building block. We
develop a diagrammatic rule for expanding the reduced Pfaffian. Then by
organizing diagrams in appropriate groups and applying the cross-ratio
identities, we show that all potential contributions to higher-order poles in
the reduced Pfaffian are canceled out, i.e., only single poles survive in
Yang-Mills theory and gravity. Furthermore, we show the cancelations of
higher-order poles in other field theories by introducing appropriate
truncations, based on the single pole structure of Pfaffian.Comment: 30 pages,6 figures,1 table, footnote adde
Note on symmetric BCJ numerator
We present an algorithm that leads to BCJ numerators satisfying manifestly
the three properties proposed by Broedel and Carrasco in [35]. We explicitly
calculate the numerators at 4, 5 and 6-points and show that the relabeling
property is generically satisfied.Comment: 14 pages, typo in eq.(4.1)is correcte
Dual-color decompositions at one-loop level in Yang-Mills theory
In this work, we extend the construction of dual color decomposition in
Yang-Mills theory to one-loop level, i.e., we show how to write one-loop
integrands in Yang-Mills theory to the dual DDM-form and the dual trace-form.
In dual forms, integrands are decomposed in terms of color-ordered one-loop
integrands for color scalar theory with proper dual color coefficients.In dual
DDM decomposition, The dual color coefficients can be obtained directly from
BCJ-form by applying Jacobi-like identities for kinematic factors. In dual
trace decomposition, the dual trace factors can be obtained by imposing
one-loop KK relations, reflection relation and their relation with the
kinematic factors in dual DDM-form.Comment: 26 pages,5 figure
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