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Table Of Integrals Series And Products

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Federated learning (FL) is a machine learning setting where many clients (e.g. mobile devices or whole organizations) collaboratively train a model under the orchestration of a central server (e.g. service provider), while keeping the training data decentralized. FL embodies the principles of focused data collection and minimization, and can mitigate many of the systemic privacy risks and costs resulting from traditional, centralized machine learning and data science approaches. Motivated by the explosive growth in FL research, this paper discusses recent advances and presents an extensive collection of open problems and challenges.

Advances and Open Problems in Federated Learning

The Bootstrap and Edgeworth Expansion

We propose a general methodology for the construction and analysis of minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the alphabet size $S$ is unknown and may be comparable with the number of observations $n$. We treat the respective regions where the functional is "nonsmooth" and "smooth" separately. In the "nonsmooth" regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the "smooth" regime, we apply a bias-corrected Maximum Likelihood Estimator (MLE). We illustrate the merit of this approach by thoroughly analyzing two important cases: the entropy $H(P) = \sum_{i = 1}^S -p_i \ln p_i$ and $F_\alpha(P) = \sum_{i = 1}^S p_i^\alpha,\alpha>0$. We obtain the minimax $L_2$ rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity $n \asymp S/\ln S$ for entropy estimation. We also show that the sample complexity for estimating $F_\alpha(P),0<\alpha<1$ is $n\asymp S^{1/\alpha}/ \ln S$, which can be achieved by our estimator but not the MLE. For $1<\alpha<3/2$, we show the minimax $L_2$ rate for estimating $F_\alpha(P)$ is $(n\ln n)^{-2(\alpha-1)}$ regardless of the alphabet size, while the $L_2$ rate for the MLE is $n^{-2(\alpha-1)}$. For all the above cases, the behavior of the minimax rate-optimal estimators with $n$ samples is essentially that of the MLE with $n\ln n$ samples. We highlight the practical advantages of our schemes for entropy and mutual information estimation. We demonstrate that our approach reduces running time and boosts the accuracy compared to existing various approaches. Moreover, we show that the mutual information estimator induced by our methodology leads to significant performance boosts over the Chow--Liu algorithm in learning graphical models.

Minimax Estimation of Functionals of Discrete Distributions

We propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size $S$ is unknown and may be comparable with or even much larger than the number of observations $n$ . We treat the respective regions where the functional is nonsmooth and smooth separately. In the nonsmooth regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the smooth regime, we apply a bias-corrected version of the maximum likelihood estimator (MLE). We illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures: 1) the entropy $H(P) = \sum _{i = 1}^{S} -p_{i} \ln p_{i}$ and 2) $F_\alpha (P) = \sum _{i = 1}^{S} p_{i}^\alpha ,\alpha >0$ . We obtain the minimax $L_{2}$ rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity $n \asymp S/\ln S$ for entropy estimation. We also demonstrate that the sample complexity for estimating $F_\alpha (P),0 , is $n\asymp S^{1/\alpha }/ \ln S$ , which can be achieved by our estimator but not the MLE. For $1 , we show the minimax $L_{2}$ rate for estimating $F_\alpha (P)$ is $(n\ln n)^{-2(\alpha -1)}$ for infinite support size, while the maximum $L_{2}$ rate for the MLE is $n^{-2(\alpha -1^{^{^{}}})}$ . For all the above cases, the behavior of the minimax rate-optimal estimators with $n$ samples is essentially that of the MLE (plug-in rule) with $n\ln n$ samples, which we term “effective sample size enlargement.” We highlight the practical advantages of our schemes for the estimation of entropy and mutual information. We compare our performance with various existing approaches, and demonstrate that our approach reduces running time and boosts the accuracy. Moreover, we show that the minimax rate-optimal mutual information estimator yielded by our framework leads to significant performance boosts over the Chow–Liu algorithm in learning graphical models. The wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable.

Location-aware networks are of great importance and interest in both civil and military applications. This paper determines the localization accuracy of an agent, which is equipped with an antenna array and localizes itself using wireless measurements with anchor nodes, in a far-field environment. In view of the Cramer–Rao bound, we first derive the localization information for static scenarios and demonstrate that such information is a weighed sum of Fisher information matrices from each anchor-antenna measurement pair. Each matrix can be further decomposed into two parts: 1) a distance part with intensity proportional to the squared baseband effective bandwidth of the transmitted signal and 2) a direction part with intensity associated with the normalized anchor-antenna visual angle. Moreover, in dynamic scenarios, we show that the Doppler shift contributes additional direction information, with intensity determined by the agent velocity and the root mean squared time duration of the transmitted signal. In addition, two measures are proposed to evaluate the localization performance of wireless networks with different anchor-agent and array-antenna geometries, and both formulae and simulations are provided for typical anchor deployments and antenna arrays.

Performance Limits and Geometric Properties of Array Localization

We consider the problem of estimating functionals of discrete distributions, and focus on a tight (up to universal multiplicative constants for each specific functional) nonasymptotic analysis of the worst case squared error risk of widely used estimators. We apply concentration inequalities to analyze the random fluctuation of these estimators around their expectations and the theory of approximation using positive linear operators to analyze the deviation of their expectations from the true functional, namely their bias . We explicitly characterize the worst case squared error risk incurred by the maximum likelihood estimator (MLE) in estimating the Shannon entropy $H(P) = \sum _{i = 1}^{S} -p_{i} \ln p_{i}$ , and the power sum $F_\alpha (P) = \sum _{i = 1}^{S} p_{i}^\alpha ,\alpha >0$ , up to universal multiplicative constants for each fixed functional, for any alphabet size $S\leq \infty $ and sample size $n$ for which the risk may vanish. As a corollary, for Shannon entropy estimation, we show that it is necessary and sufficient to have $n \gg S$ observations for the MLE to be consistent. In addition, we establish that it is necessary and sufficient to consider $n \gg S^{1/\alpha }$ samples for the MLE to consistently estimate $F_\alpha (P), 0 . The minimax rate-optimal estimators for both problems require $S/\ln S$ and $S^{1/\alpha }/\ln S$ samples, which implies that the MLE has a strictly sub-optimal sample complexity. When $1 , we show that the worst case squared error rate of convergence for the MLE is $n^{-2(\alpha -1)}$ for infinite alphabet size, while the minimax squared error rate is $(n\ln n)^{-2(\alpha -1)}$ . When $\alpha \geq 3/2$ , the MLE achieves the minimax optimal rate $n^{-1}$ regardless of the alphabet size. As an application of the general theory, we analyze the Dirichlet prior smoothing techniques for Shannon entropy estimation. In this context, one approach is to plug-in the Dirichlet prior smoothed distribution into the entropy functional, while the other one is to calculate the Bayes estimator for entropy under the Dirichlet prior for squared error, which is the conditional expectation. We show that in general such estimators do not improve over the maximum likelihood estimator. No matter how we tune the parameters in the Dirichlet prior, this approach cannot achieve the minimax rates in entropy estimation. The performance of the minimax rate-optimal estimator with $n$ samples is essentially at least as good as that of Dirichlet smoothed entropy estimators with $n\ln n$ samples.

Maximum Likelihood Estimation of Functionals of Discrete Distributions

Location-aware networks are of great importance and interest in both civil and military applications. This paper determines the localization accuracy of an agent, which is equipped with an antenna array and localizes itself using wireless measurements with anchor nodes, in a far-field environment. In view of the Cramer-Rao bound, we first derive the localization information for static scenarios and demonstrate that such information is a weighed sum of Fisher information matrices from each anchor-antenna measurement pair. Each matrix can be further decomposed into two parts: a distance part with intensity proportional to the squared baseband effective bandwidth of the transmitted signal and a direction part with intensity associated with the normalized anchor-antenna visual angle. Moreover, in dynamic scenarios, we show that the Doppler shift contributes additional direction information, with intensity determined by the agent velocity and the root mean squared time duration of the transmitted signal. In addition, two measures are proposed to evaluate the localization performance of wireless networks with different anchor-agent and array-antenna geometries, and both formulae and simulations are provided for typical anchor deployments and antenna arrays.

Yanjun Han

Papers

Minimax Estimation of Functionals of Discrete Distributions

Minimax Estimation of Functionals of Discrete Distributions

Performance Limits and Geometric Properties of Array Localization

Maximum Likelihood Estimation of Functionals of Discrete Distributions

Performance Limits and Geometric Properties of Array Localization