Algorithms for Sparse Non-negative Tucker Decompositions

There is a increasing interest in analysis of large scale multi-way data. The concept of multi-way data refers to arrays of data with more than two dimensions, i.e., taking the form of tensors. To analyze such data, decomposition techniques are widely used. The two most common decompositions for tensors are the Tucker model and the more restricted PARAFAC model. Both models can be viewed as generalizations of the regular factor analysis to data of more than two modalities. Non-negative matrix factorization (NMF) in conjunction with sparse coding has lately been given much attention due to its part based and easy interpretable representation. While NMF has been extended to the PARAFAC model no such attempt has been done to extend NMF to the Tucker model. However, if the tensor data analyzed is non-negative it may well be relevant to consider purely additive (i.e., non-negative Tucker decompositions). To reduce ambiguities of this type of decomposition we develop updates that can impose sparseness in any combination of modalities, hence, proposed algorithms for sparse non-negative Tucker decompositions (SN-TUCKER). We demonstrate how the proposed algorithms are superior to existing algorithms for Tucker decompositions when indeed the data and interactions can be considered non-negative. We further illustrate how sparse coding can help identify what model (PARAFAC or Tucker) is the most appropriate for the data as well as to select the number of components by turning off excess components. The algorithms for SN-TUCKER can be downloaded from www.erpwavelab.org.