instruction:
In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
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This manuscript presents a trust computation for international relations and its calculus, which related to Bayesian inference, Dempster-Shafer theory and subjective logic. We proposed a method that allows a trust computation which is previously subjective and incomputable. An example of case study for the trust computation is the United States of America–Great Britain relations. The method supports decision makers in a government such as foreign ministry, defense ministry, presidential or prime minister office. The Department of Defense (DoD) may use our method to determine a nation that can be known as a friendly, neutral or hostile nation.
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A Mathematical Trust Algebra for International Nation Relations Computation and Evaluation


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We consider linear models for stochastic dynamics. To any such model can be as-<lb>sociated a network (namely a directed graph) describing which degrees of freedom<lb>interact under the dynamics. We tackle the problem of learning such a network<lb>from observation of the system trajectory over a time interval T .<lb>We analyze the l1-regularized least squares algorithm and, in the setting in which<lb>the underlying network is sparse, we prove performance guarantees that are uni-<lb>form in the sampling rate as long as this is sufficiently high. This result substan-<lb>tiates the notion of a well defined ‘time complexity’ for the network inference<lb>problem. keywords: Gaussian processes, model selection and structure learning, graphical models, sparsity<lb>and feature selection.
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Learning Networks of Stochastic Differential Equations


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Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degreeO(poly log(1/ )) which is -close, and similarly for any rational function there exists a ReLU network of size O(poly log(1/ )) which is -close. By contrast, polynomials need degree Ω(poly(1/ )) to approximate even a single ReLU. When converting a ReLU network to a rational function as above, the hidden constants depend exponentially on the number of layers, which is shown to be tight; in other words, a compositional representation can be beneficial even for rational functions. 1. Overview Significant effort has been invested in characterizing the functions that can be efficiently approximated by neural networks. The goal of the present work is to characterize neural networks more finely by finding a class of functions which is not only well-approximated by neural networks, but also well-approximates neural networks. The function class investigated here is the class of rational functions: functions represented as the ratio of two polynomials, where the denominator is a strictly positive polynomial. For simplicity, the neural networks are taken to always use ReLU activation σr(x) := max{0, x}; for a review of neural networks and their terminology, the reader is directed to Section 1.4. For the sake of brevity, a network with ReLU activations is simply called a ReLU network. 1.1. Main results The main theorem here states that ReLU networks and rational functions approximate each other well in the sense University of Illinois, Urbana-Champaign; work completed while visiting the Simons Institute. Correspondence to: your friend <mjt@illinois.edu>. Proceedings of the 34 th International Conference on Machine Learning, Sydney, Australia, PMLR 70, 2017. Copyright 2017 by the author(s). −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 0 1 2 3 4 spike rat poly net Figure 1. Rational, polynomial, and ReLU network fit to “spike”, a function which is 1/x along [1/4, 1] and 0 elsewhere. that -approximating one class with the other requires a representation whose size is polynomial in ln(1 / ), rather than being polynomial in 1/ . Theorem 1.1. 1. Let ∈ (0, 1] and nonnegative integer k be given. Let p : [0, 1] → [−1,+1] and q : [0, 1] → [2−k, 1] be polynomials of degree ≤ r, each with≤ smonomials. Then there exists a function f : [0, 1] → R, representable as a ReLU network of size (number of nodes) O ( k ln(1 / ) + min { srk ln(sr / ), sdk ln(dsr / ) }) ,
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Neural networks and rational functions