The Blahut-Arimoto (BA) algorithm has played a fundamental role in the numerical computation of rate-distortion (RD) functions. This algorithm possesses a desirable monotonic convergence property by alternatively minimizing its Lagrangian with a fixed multiplier. In this paper, we propose a novel modification of the BA algorithm, letting the multiplier be updated in each iteration via a one-dimensional root-finding step with respect to a monotonic univariate function, which can be efficiently implemented by Newton's method. This allows the multiplier to be updated in a flexible and efficient manner, overcoming a major drawback of the original BA algorithm wherein the multiplier is fixed throughout iterations. Consequently, the modified algorithm is capable of directly computing the RD function for a given target distortion, without exploring the entire RD curve as in the original BA algorithm. A theoretical analysis shows that the modified algorithm still converges to the RD function and the convergence rate is $\Theta(1/n)$, where $n$ denotes the number of iterations. Numerical experiments demonstrate that the modified algorithm directly computes the RD function with a given target distortion, and it significantly accelerates the original BA algorithm.

翻译：Blahut-Arimoto(BA)算法在率失真函数的数值计算中发挥着基础性作用。该算法通过交替最小化具有固定乘子的拉格朗日函数，具有理想的单调收敛性质。本文提出一种新颖的BA算法改进方案，每次迭代中通过关于单调单变量函数的一维求根步骤更新乘子，该过程可采用牛顿法高效实现。这种设计使得乘子能够以灵活高效的方式更新，克服了原始BA算法中乘子在迭代过程中保持固定的主要缺陷。改进算法能够直接计算给定目标失真下的率失真函数，无需像原始BA算法那样探索整条率失真曲线。理论分析表明，改进算法仍收敛于率失真函数，且收敛速率为$\Theta(1/n)$，其中$n$表示迭代次数。数值实验证明，改进算法能够直接计算给定目标失真下的率失真函数，并显著加速原始BA算法的运行。