Sparse Coding and Automatic Relevance Determination for Multi-way models

Multi-way modeling has become an important tool in the analysis of large scale multi-modal data. An important class of multi-way models is given by the Tucker model which decomposes the data into components pertaining to each modality as well as a core array indicating how the components of the various modalities interact. Unfortunately, the Tucker model is not unique. Furthermore, establishing the adequate model order is difficult as the number of components are specified for each mode separately. Previously, rotation criteria such as VARIMAX has been used to resolve the non-uniqueness of the Tucker representation [7]. Furthermore, all potential models have been exhaustively evaluated to estimate the adequate number of components of each mode. We demonstrate how sparse coding can prune excess components and resolve the non-uniqueness of the Tucker model while Automatic Relevance Determination in Bayesian learning form a framework to learn the adequate degree of sparsity imposed. On a wide range of multi-way data sets the proposed method is demonstrated to successfully prune excess components thereby establishing the model order. Furthermore, the non-uniqueness of the Tucker model is resolved since among potential models the models giving the sparsest representation as measured by the sparse coding regularization is attained. The approach readily generalizes to regular sparse coding as well as the CandeComp/PARAFAC model as both models are special cases of the Tucker model.