Automatic relevance determination for multi‐way models

Estimating the adequate number of components is an important yet difficult problem in multi‐way modeling. We demonstrate how a Bayesian framework for model selection based on automatic relevance determination (ARD) can be adapted to the Tucker and CandeComp/PARAFAC (CP) models. By assigning priors for the model parameters and learning the hyperparameters of these priors the method is able to turn off excess components and simplify the core structure at a computational cost of fitting the conventional Tucker/CP model. To investigate the impact of the choice of priors we based the ARD on both Laplace and Gaussian priors corresponding to regularization by the sparsity promoting l1‐norm and the conventional l2‐norm, respectively. While the form of the priors had limited effect on the results obtained the ARD approach turned out to form a useful, simple, and efficient tool for selecting the adequate number of components of data within the Tucker and CP structure. For the Tucker and CP model the approach performs better than heuristics such as the Bayesian information criterion (BIC), Akaikes information criterion (AIC), DIFFIT and the numerical convex hull (NumConvHull) while operating only at the cost of estimating an ordinary CP/Tucker model. For the CP model the ARD approach performs almost as well as the core consistency diagnostic (CorConDiag). Thus, the ARD framework is a simple yet efficient tool for the estimation of the adequate number of components in multi‐way models. A Matlab implementation of the proposed algorithm is available for download at www.erpwavelab.org. Copyright © 2009 John Wiley & Sons, Ltd.