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, wherein the multiplier is updated through a one-dimensional root-finding step using a monotonic univariate function, efficiently implemented by Newton's method in each iteration. Consequently, the modified algorithm directly computes the RD function for a given target distortion, without exploring the entire RD curve as in the original BA algorithm. Moreover, this modification presents a versatile framework, applicable to a wide range of problems, including the computation of distortion-rate (DR) functions. Theoretical analysis shows that the outputs of the modified algorithms still converge to the solutions of the RD and DR functions with rate $O(1/n)$, where $n$ is the number of iterations. Additionally, these algorithms provide $\varepsilon$-approximation solutions with $O\left(\frac{MN\log N}{\varepsilon}(1+\log |\log \varepsilon|)\right)$ arithmetic operations, where $M,N$ are the sizes of source and reproduced alphabets respectively. Numerical experiments demonstrate that the modified algorithms exhibit significant acceleration compared with the original BA algorithms and showcase commendable performance across classical source distributions such as discretized Gaussian, Laplacian and uniform sources.

翻译：Blahut-Arimoto (BA)算法在率失真(RD)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固定乘子的拉格朗日函数，具备理想的单调收敛特性。本文提出一种新颖的BA算法改进方案，其中乘子通过利用单调单变量函数的一维求根步骤进行更新，并在每次迭代中采用牛顿法高效实现。改进算法无需像原始BA算法那样探索完整RD曲线，即可直接计算给定目标失真下的RD函数。此外，该改进方案呈现出一个通用框架，适用于包括失真率(DR)函数计算在内的广泛问题。理论分析表明，改进算法的输出仍以$O(1/n)$的速率收敛至RD和DR函数的解，其中$n$为迭代次数。同时，这些算法通过$O\left(\frac{MN\log N}{\varepsilon}(1+\log |\log \varepsilon|)\right)$次算术运算提供$\varepsilon$近似解，其中$M,N$分别为信源字母集与重建字母集的大小。数值实验表明，与原始BA算法相比，改进算法展现出显著加速效果，并在离散化高斯、拉普拉斯和均匀信源等经典信源分布上表现出优异的性能。