Efficient Computation of the PARAFAC2 Decomposition via Generalized Tensor Contractions

The PARAFAC2 decomposition is a generalization of the PARAFAC/CP (Canonical Polyadic) tensor decomposition. This tensor decomposition represents a set of coupled matrix decompositions with one mode in common, i.e., one of the components varies along the set of matrices (tensor slices), whereas the second component stays constant. It has many applications including the analysis of chromatographic data, data clustering, and biomedical signal processing. Typically, the PARAFAC2 decomposition is described using a slice-wise notation, where each of the slices corresponds to one of the matrix decompositions. In this paper, we first present a generalization of the slice-wise multiplication based on the tensor contraction operator, where the generalized contraction operator represents an inner product between two tensors of compatible dimensions. Next, we express the PARAFAC2 decomposition in terms of this new tensor model resulting in a constrained CP model. Moreover, we show that this new description opens an efficient single loop way to compute the PARAFAC2 decomposition based on the generalized tensor contraction.