Flexible and efficient implementations of Bayesian independent component analysis

Abstract

In this paper we present an empirical Bayes method for flexible and efficient Independent Component Analysis (ICA). The method is flexible with respect to choice of source prior, dimensionality and positivity of the mixing matrix, and structure of the noise covariance matrix. The efficiency is ensured using parameter optimizers which are more advanced than the expectation maximization (EM) algorithm, but still easy to implement. These optimizers are the overrelaxed adaptive EM algorithm and the easy gradient recipe. The required expectations over the source posterior are estimated with accurate mean field methods: variational and the expectation consistent framework.We demonstrate the usefulness of the approach with the publicly available Matlab toolbox icaMF. Flexible and Efficient Implementations of Bayesian Independent Component Analysis AbstractIn this paper we present an empirical Bayes method for flexible and efficient Independent Component Analysis (ICA). The method is flexible with respect to choice of source prior, dimensionality and constraints of the mixing matrix (positivity), and structure of the noise covariance matrix. The efficiency is ensured using parameter optimizers which are more advanced than the expectation maximization (EM) algorithm, but still easy to implement. These optimizers are the overrelaxed adaptive EM algorithm and the easy gradient recipe. The required expectations over the source posterior are estimated with mean field methods: variational and the expectation consistent (EC) framework. We describe the derivation of the EC framework for ICA in detail and give empirical results demonstrating the improved performance. We demonstrate the usefulness of the approach with the publicly available Matlab toolbox icaMF.

Bayesian independent component analysis: Variational methods and non-negative decompositions

“…A consequence of using the factorized variational distribution is that we will make trivial predictions for the non-diagonal second moments: s i s i q = s i q s i q for i = i . Although these correlations do not appear in the variational updates they will play an important and sometimes crucial role parameter estimation [20]. Fortunately, the linear response correction [5] and expectation consistent (EC) framework [16] give non-trivial estimates of correlations.…”

“…Secondly, an empirical comparison in Ref. [20] shows that EC gives more precise results both in estimating source moments and in solving the ICA problem.…”

“…An important remaining question is how to tune the parameters to actually fulfill the expectation consistent conditions. In [20] we give a recipe based upon Tom Minka's expectation propagation framework (EP) [9]. We also give explicit expressions for the marginal likelihood that we repeat here for completeness.…”

“…This make a very significant difference since the q distribution is factorized whereas r is a multivariate Gaussian. Special care has to be taken to constrained variables [20], e.g. below we discuss multiplicative updates for non-negative A.…”

“…An obvious question is how much difference in practical situations there will be between the different methods. In our companion paper [20] we have made an effort to answer this question. Although simulations can never be exhaustive they support the following conclusions: The best results are obtained using the EC framework and the advanced optimization methods.…”