文章基本信息

标题：Multivariate sharp quadratic bounds via $\mathbf{\Sigma}$-strong convexity and the Fenchel connection
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作者：Ryan P. Browne ; Paul D. McNicholas
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2015
卷号：9
期号：2
页码：1913-1938
DOI：10.1214/15-EJS1061
语种：English
出版社：Institute of Mathematical Statistics
摘要：Sharp majorization is extended to the multivariate case. To achieve this, the notions of $\sigma$-strong convexity, monotonicity, and one-sided Lipschitz continuity are extended to $\mathbf{\Sigma}$-strong convexity, monotonicity, and Lipschitz continuity, respectively. The connection between a convex function and its Fenchel-Legendre transform is then developed. Sharp majorization is illustrated in single and multiple dimensions, and we show that these extensions yield improvements on bounds given within the literature. The new methodology introduced herein is used to develop a variational approximation for the Bayesian multinomial regression model.