A Bounded Measure for Estimating the Benefit of Visualization (Part I): Theoretical Discourse and Conceptual Evaluation

Information theory can be used to analyze the cost–benefit of visualization processes. However, the current measure of benefit contains an unbounded term that is neither easy to estimate nor intuitive to interpret. In this work, we propose to revise the existing cost–benefit measure by replacing the unbounded term with a bounded one. We examine a number of bounded measures that include the Jenson–Shannon divergence, its square root, and a new divergence measure formulated as part of this work. We describe the rationale for proposing a new divergence measure. In the first part of this paper, we focus on the conceptual analysis of the mathematical properties of these candidate measures. We use visualization to support the multi-criteria comparison, narrowing the search down to several options with better mathematical properties. The theoretical discourse and conceptual evaluation in this part provides the basis for further data-driven evaluation based on synthetic and experimental case studies that are reported in the second part of this paper.     

一种单位化的增量梯度算法

研究目标函数是若干光滑函数和的可分离优化问题,提出了一种单位化增量梯度算法.该算法每次子迭代只需要计算一个(或几个)分量函数的单位负梯度方向作为迭代方向.在一定条件下,证明了采用发散步长的单位化增量梯度算法的收敛性.作为应用,新算法和Bertsekas D P,Tsitsikils J N提出的(没有单位化)增量梯度算...     

关于一道高等数学竞赛试题的探索与拓广

给出了2004年浙江省大学生高等数学竞赛一题得分率较低的压轴题(判断级数sum from n=1 to ∞ 1/n((n!)~α)～(1/n)的敛散性,其中α>0为常数)的五种不同的解法,建立了它的如下的拓广结果:当α>1且正项级数sum from i=1 to ∞ 1/(a_i~α)收敛时,级数sum from n=1 to ∞ 1/((multiply from i=1 to n)ai)α～(1/n)收敛;当0<α≤1,0相似文献   

一种新的梯形模糊数间距离及在投资组合模型选择中的应用

给出一种新的模糊数间的距离公式.用新定义的距离公式来度量给定优先资产的条件下投资组合的分散度.用可能性均值度量投资组合的收益,可能性半方差度量投资组合的风险,在收益和风险满足一定的条件下构造分散度模型.通过实例分析,给出的方法不仅分散度更好,而且资产分配多元化程度更高,计算也更简单.     

Error Exponents and α-Mutual Information

Over the last six decades, the representation of error exponent functions for data transmission through noisy channels at rates below capacity has seen three distinct approaches: (1) Through Gallager’s E0 functions (with and without cost constraints); (2) large deviations form, in terms of conditional relative entropy and mutual information; (3) through the α-mutual information and the Augustin–Csiszár mutual information of order α derived from the Rényi divergence. While a fairly complete picture has emerged in the absence of cost constraints, there have remained gaps in the interrelationships between the three approaches in the general case of cost-constrained encoding. Furthermore, no systematic approach has been proposed to solve the attendant optimization problems by exploiting the specific structure of the information functions. This paper closes those gaps and proposes a simple method to maximize Augustin–Csiszár mutual information of order α under cost constraints by means of the maximization of the α-mutual information subject to an exponential average constraint.     

Re-examination of Bregman functions and new properties of their divergences

ABSTRACT

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the ‘Bregman function’). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charvát-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have ‘Bregmanian properties’, can, by our re-examination of the theory, be considered as a Bregman function. Our analysis yields several by-products of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.     

对推广Raabe判别法的再讨论

推广Raabe判别法是新近提出的关于正项级数敛散性问题一种普遍性方法.通过对它的进一步探讨,推出了几种常用判别法,同时得到了推广Raade判别法与经典的Kummer判别法的关系.     

一种基于电势分布的电容成像敏感分布计算方法

系统分析了电容成像(Electrical capacitance tomography,简称ECT)系统敏感分布和电势分布的关系,统一了计算公式,该目的简单且计算量小.仿真表明,采用该目的计算的ECT敏感分布,与采用定义计算的结果吻合.