Sparse Tucker2 analysis of three-way data subject to a constrained number of zero elements in a core array

Abstract

Three-way principal component analysis (3WPCA) models have been developed for analyzing a three-way data array of objects × variables × sources. Among the 3WPCA models, the least restrictive is the Tucker2 model, in which an extended core array describes the source-specific relationships between the components underlying objects and those for variables. In contrast to Tucker2 with the core array unconstrained, the Parafac model is highly restrictive in that the core slices in the array are constrained to be diagonal matrices. In this paper, we propose a procedure by which a suitably constrained intermediate model between Tucker2 and Parafac can be found. In the proposed procedure, the Tucker2 loss function is minimized subject to a specified number of core elements being zero with their locations unknown; the optimal locations of zero elements and nonzero parameter values are simultaneously estimated. This technique can be called sparse core Tucker2 (ScTucker2), as the matrices including a number of zeros are said to be sparse. We present an alternating least squares algorithm for ScTucker2 with a procedure for selecting a suitable number of zero elements. This procedure is assessed in a simulation study and illustrated with real data sets.