A C C Coolen

Alberto Fachechi, Elena Agliari, Miriam Aquaro et al.

We consider restricted Boltzmann machines with a binary visible layer and a Gaussian hidden layer trained by an unlabelled dataset composed of noisy realizations of a single ground pattern. We develop a statistical mechanics framework to describe the network generative capabilities, by exploiting the replica trick and assuming self-averaging of the underlying order parameters (i.e., replica symmetry). In particular, we outline the effective control parameters (e.g., the relative number of weights to be trained, the regularization parameter), whose tuning can yield qualitatively-different operative regimes. Further, we provide analytical and numerical evidence for the existence of a sub-region in the space of the hyperparameters where replica-symmetry breaking occurs.

M Sheikh, A C C Coolen

We extend the standard Bayesian multivariate Gaussian generative data classifier by considering a generalization of the conjugate, normal-Wishart prior distribution and by deriving the hyperparameters analytically via evidence maximization. The behaviour of the optimal hyperparameters is explored in the high-dimensional data regime. The classification accuracy of the resulting generalized model is competitive with state-of-the art Bayesian discriminant analysis methods, but without the usual computational burden of cross-validation.

James Barrett, Anthony C. C. Coolen

The Gaussian Process Latent Variable Model (GP-LVM) is a non-linear probabilistic method of embedding a high dimensional dataset in terms low dimensional `latent' variables. In this paper we illustrate that maximum a posteriori (MAP) estimation of the latent variables and hyperparameters can be used for model selection and hence we can determine the optimal number or latent variables and the most appropriate model. This is an alternative to the variational approaches developed recently and may be useful when we want to use a non-Gaussian prior or kernel functions that don't have automatic relevance determination (ARD) parameters. Using a second order expansion of the latent variable posterior we can marginalise the latent variables and obtain an estimate for the hyperparameter posterior. Secondly, we use the GP-LVM to integrate multiple data sources by simultaneously embedding them in terms of common latent variables. We present results from synthetic data to illustrate the successful detection and retrieval of low dimensional structure from high dimensional data. We demonstrate that the integration of multiple data sources leads to more robust performance. Finally, we show that when the data are used for binary classification tasks we can attain a significant gain in prediction accuracy when the low dimensional representation is used.