Blahut–Arimoto algorithm

For the case of channel capacity, the algorithm was independently invented by Suguru Arimoto[1] and Richard Blahut.[2] In the case of lossy compression, the corresponding algorithm was invented by Richard Blahut. The algorithm is most applicable to the case of arbitrary finite alphabet sources. Much work has been done to extend it to more general problem instances.[3][4] Recently, a version of the algorithm that accounts for continuous and multivariate outputs was proposed with applications in cellular signaling [5]. There exists also a version of Blahut–Arimoto algorithm for directed information[6].

Suppose we have a source ${\displaystyle X}$ with probability ${\displaystyle p(x)}$ of any given symbol. We wish to find an encoding ${\displaystyle p({\hat {x}}|x)}$ that generates a compressed signal ${\displaystyle {\hat {X}}}$ from the original signal while minimizing the expected distortion ${\displaystyle \langle d(x,{\hat {x}})\rangle }$, where the expectation is taken over the joint probability of ${\displaystyle X}$ and ${\displaystyle {\hat {X}}}$. We can find an encoding that minimizes the rate-distortion functional locally by repeating the following iteration until convergence:

${\displaystyle p_{t+1}({\hat {x}})=\sum _{x}p(x)p_{t}({\hat {x}}|x)}$

${\displaystyle p_{t+1}({\hat {x}}|x)={\frac {p_{t}({\hat {x}})\exp(-\beta d(x,{\hat {x}}))}{\sum _{\hat {x}}p_{t}({\hat {x}})\exp(-\beta d(x,{\hat {x}}))}}}$

where ${\displaystyle \beta }$ is a parameter related to the slope in the rate-distortion curve that we are targeting and thus is related to how much we favor compression versus distortion (higher ${\displaystyle \beta }$ means less compression).