A Bayesian ensemble learning method is introduced for unsupervised extraction of dynamic processes from noisy data. The data are assumed to be generated by an unknow n nonlinear mapping from unknow n factors. The dynamics of the factors are modeled using a nonlinear state-space model. The nonlinear mappings in the model are represented using multilayer perceptron networks. The proposed method is computationally demanding, but it allows the use of higher-dimensional nonlinear latent variable models than other existing approaches. Experiments with chaotic data show that the new method is able to blindly estimate the factors and the dynamic process that generated the data. It clearly outperforms currently available nonlinear prediction techniques in this very dif cult test problem.

A network supporting deep unsupervised learning is presented. The network is an autoencoder with lateral shortcut connections from the encoder to decoder at each level of the hierarchy. The lateral shortcut connections allow the higher levels of the hierarchy to focus on abstract invariant features. While standard autoencoders are analogous to latent variable models with a single layer of stochastic variables, the proposed network is analogous to hierarchical latent variables models. Learning combines denoising autoencoder and denoising sources separation frameworks. Each layer of the network contributes to the cost function a term which measures the distance of the representations produced by the encoder and the decoder. Since training signals originate from all levels of the network, all layers can learn efficiently even in deep networks. The speedup offered by cost terms from higher levels of the hierarchy and the ability to learn invariant features are demonstrated in experiments.

The bits-back coding first introduced by Wallace in 1990 and later by Hinton and van Camp in 1993 provides an interesting link between Bayesian learning and information-theoretic minimum-description-length (MDL) learning approaches. The bits-back coding allows interpreting the cost function used in the variational Bayesian method called ensemble learning as a code length in addition to the Bayesian view of misfit of the posterior approximation and a lower bound of model evidence. Combining these two viewpoints provides interesting insights to the learning process and the functions of different parts of the model. In this paper, the problem of variational Bayesian learning of hierarchical latent variable models is used to demonstrate the benefits of the two views. The code-length interpretation provides new views to many parts of the problem such as model comparison and pruning and helps explain many phenomena occurring in learning.

In this paper, we introduce a modeling approach called independent variable group analysis (IVGA) which can be used for finding an efficient structural representation for a given data set. The basic idea is to determine such a grouping for the variables of the data set that mutually dependent variables are grouped together whereas mutually independent or weakly dependent variables end up in separate groups. Computation of an IVGA model requires a combinatorial algorithm for grouping of the variables and a modeling algorithm for the groups. In order to be able to compare different groupings, a cost function which reflects the quality of a grouping is also required. Such a cost function can be derived, for example, using the variational Bayesian approach, which is employed in our study. This approach is also shown to be approximately equivalent to minimizing the mutual information between the groups. The modeling task is computationally demanding. We describe an efficient heuristic grouping algorithm for the variables and derive a computationally light nonlinear mixture model for modeling of the dependencies within the groups. Finally, we carry out a set of experiments which indicate that IVGA may turn out to be beneficial in many different applications.Index Terms-Compact modeling, independent variable group analysis (IVGA), mutual information, variable grouping, variational Bayesian learning.

In many models, variances are assumed to be constant although this assumption is often unrealistic in practice. Joint modelling of means and variances is difficult in many learning approaches, because it can lead into infinite probability densities. We show that a Bayesian variational technique which is sensitive to probability mass instead of density is able to jointly model both variances and means. We consider a model structure where a Gaussian variable, called variance node, controls the variance of another Gaussian variable. Variance nodes make it possible to build hierarchical models for both variances and means. We report experiments with artificial data which demonstrate the ability of the learning algorithm to find variance sources explaining and characterizing well the variances in the multidimensional data. Experiments with biomedical MEG data show that variance sources are present in real-world signals.

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We show that the choice of posterior approximation of sources affects the solution found in Bayesian variational learning of linear independent component analysis models. Assuming the sources to be independent a posteriori favours a solution which has an orthogonal mixing matrix. A linear dynamic model which uses second-order statistics is considered but the analysis extends to nonlinear mixtures and non-Gaussian source models as well.

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