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Volume 6, Issue 2
A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions

Lingyi Chen, Shitong Wu, Wenhao Ye, Huihui Wu, Wenyi Zhang, Hao Wu & Bo Bai

CSIAM Trans. Appl. Math., 6 (2025), pp. 350-379.

Published online: 2025-05

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  • Abstract

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 $\mathcal{O}(1/n),$ where $n$ is the number of iterations. Additionally, these algorithms provide $ε$-approximation solutions with $\mathcal{O}((MN{\rm {\rm log}}N/ε)(1+{\rm log}|{\rm log}ε|))$ 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.

  • AMS Subject Headings

90C25, 94A29, 94A34

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-6-350, author = {Chen , LingyiWu , ShitongYe , WenhaoWu , HuihuiZhang , WenyiWu , Hao and Bai , Bo}, title = {A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2025}, volume = {6}, number = {2}, pages = {350--379}, abstract = {

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 $\mathcal{O}(1/n),$ where $n$ is the number of iterations. Additionally, these algorithms provide $ε$-approximation solutions with $\mathcal{O}((MN{\rm {\rm log}}N/ε)(1+{\rm log}|{\rm log}ε|))$ 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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2024-0002}, url = {http://global-sci.org/intro/article_detail/csiam-am/24089.html} }
TY - JOUR T1 - A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions AU - Chen , Lingyi AU - Wu , Shitong AU - Ye , Wenhao AU - Wu , Huihui AU - Zhang , Wenyi AU - Wu , Hao AU - Bai , Bo JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 350 EP - 379 PY - 2025 DA - 2025/05 SN - 6 DO - http://doi.org/10.4208/csiam-am.SO-2024-0002 UR - https://global-sci.org/intro/article_detail/csiam-am/24089.html KW - Alternating minimization, Blahut-Arimoto algorithm, convergence analysis, constrained optimization, rate-distortion function. AB -

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 $\mathcal{O}(1/n),$ where $n$ is the number of iterations. Additionally, these algorithms provide $ε$-approximation solutions with $\mathcal{O}((MN{\rm {\rm log}}N/ε)(1+{\rm log}|{\rm log}ε|))$ 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.

Chen , LingyiWu , ShitongYe , WenhaoWu , HuihuiZhang , WenyiWu , Hao and Bai , Bo. (2025). A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions. CSIAM Transactions on Applied Mathematics. 6 (2). 350-379. doi:10.4208/csiam-am.SO-2024-0002
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