We investigate the problem of minimizing Kullback-Leibler divergence between a linear model $Ax$ and a positive vector $b$ in different convex domains (positive orthant, $n$-dimensional box, probability simplex). Our focus is on the SMART method that employs efficient multiplicative updates. We explore the exponentiated gradient method, which can be viewed as a Bregman proximal gradient method and as a Riemannian gradient descent on the parameter manifold of a corresponding distribution of the exponential family. This dual interpretation enables us to establish connections and achieve accelerated SMART iterates while smoothly incorporating constraints. The performance of the proposed acceleration schemes is demonstrated by large-scale numerical examples.

翻译：我们研究了在多种凸域（正象限、n维盒域、概率单纯形）中，线性模型$Ax$与正向量$b$之间Kullback-Leibler散度的最小化问题。重点分析了采用高效乘性更新的SMART方法，并探讨了指数梯度法——该方法可视为Bregman邻近梯度法，亦可看作指数族对应分布的参数流形上的黎曼梯度下降。这种双重解释使我们能够建立联系，在平滑引入约束的同时实现加速的SMART迭代。通过大规模数值算例验证了所提加速方案的性能。