Abstract
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.
Highlights
With the ever increasing importance of computational tasks in statistics, the class of majorization-maximization algorithms is becoming ever more relevant [7, 10, 14, 11]
Sharp majorization has been extended to the multivariate case
The notions of σ-strong convexity, monotonicity, and one-sided Lipschitz continuity have been extended to Σ-strong convexity, monotonicity, and Lipschitz continuity, respectively
Summary
With the ever increasing importance of computational tasks in statistics, the class of majorization-maximization algorithms is becoming ever more relevant [7, 10, 14, 11]. Suppose a complicated objective function f is to be minimized This can be achieved iteratively by constructing a majorizing function g at the current solution xk and finding a new solution xk+1 by minimizing the majorization function. Such an algorithm is an MM algorithm of the majorizationminimization variety, cf [11]. The connection between majorization of f and minorization of f ∗, the Fenchel-Legendre transform of f , is illustrated. This is done both in the univariate and multivariate cases. A simulation is carried out to study the computational efficiency of our approach (Section 6), and the paper concludes with a brief discussion (Section 7)
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