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The Blahut–Arimoto algorithm, co-invented by Richard Blahut, is an elegant iterative technique for numerically obtaining rate–distortion functions of arbitrary finite input/output alphabet sources. Much work has been done to extend it to more general problem instances.

Algorithm

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

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

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

where $\beta$ is an inverse temperature parameter that controls how much we favor compression versus distortion (higher $\beta$ means less compression). It should be noted that the above algorithm only converges locally to an optimal point on the rate-distortion curve. Finding the global optimum is a computationally hard problem.

References

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