The term Blahut–Arimoto algorithm is often used to refer to a class of algorithms for computing numerically either the information theoretic capacity of a channel, the rate-distortion function of a source or a source encoding (i.e. compression to remove the redundancy).
They are iterative algorithms that eventually converge to one of the maxima of the optimization problem that is associated with these information theoretic concepts.
For the case of channel capacity, the algorithm was independently invented by Suguru Arimoto[1] and Richard Blahut.
[2] In addition, Blahut's treatment gives algorithms for computing rate distortion and generalized capacity with input contraints (i.e. the capacity-cost function, analogous to rate-distortion).
These algorithms are 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] A discrete memoryless channel (DMC) can be specified using two random variables
, and a channel law as a conditional probability distribution
The channel capacity, defined as
, indicates the maximum efficiency that a channel can communicate, in the unit of bit per use.
[7] Now if we denote the cardinality
column entry by
For the case of channel capacity, the algorithm was independently invented by Suguru Arimoto[8] and Richard Blahut.
[9] They both found the following expression for the capacity of a DMC with channel law:
{\displaystyle C=\max _{\mathbf {p} }\max _{Q}\sum _{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{ij}\log \left({\dfrac {Q_{ji}}{p_{i}}}\right)}
are maximized over the following requirements: Then upon picking a random probability distribution
, we can generate a sequence
{\displaystyle (q_{ji}^{t}):={\dfrac {p_{i}^{t}w_{ij}}{\sum _{k=1}^{n}p_{k}^{t}w_{kj}}}}
{\displaystyle p_{k}^{t+1}:={\dfrac {\prod _{j=1}^{m}(q_{jk}^{t})^{w_{kj}}}{\sum _{i=1}^{n}\prod _{j=1}^{m}(q_{ji}^{t})^{w_{ij}}}}}
Then, using the theory of optimization, specifically coordinate descent, Yeung[10] showed that the sequence indeed converges to the required maximum.
{\displaystyle \lim _{t\to \infty }\sum _{i=1}^{n}\sum _{j=1}^{m}p_{i}^{t}w_{ij}\log \left({\dfrac {Q_{ji}^{t}}{p_{i}^{t}}}\right)=C}
, the capacity can be numerically estimated up to arbitrary precision.
that generates a compressed signal
from the original signal while minimizing the expected distortion
, where the expectation is taken over the joint probability of
We can find an encoding that minimizes the rate-distortion functional locally by repeating the following iteration until convergence: where
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
means less compression).