Conversion marketing

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 $\langle d(x,{\hat {x}})\rangle$, 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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