Tucker4 model for compositions: Algorithm and Interpretation

Tensors are the preferred algebraic architecture for storing complex data and describing mul- tilinear relationships between entities in a compact form. A generic nth-order tensor stores data along n indices and can be described as a generalization of simple structures such as scalars, vectors and matrices which are special cases of 0-order (no index), 1st-order (1 index) and 2nd- order (2 indices) tensors. Tensor decompositions tools allow capturing multidimensional information by breaking it down in sets of simpler objects, generally lower order tensors. Tucker is one of most commonly used technique for decomposition of such objects. It is suitable for summarizing large information into condensed sets of variables; thus, it is the preferred method for tensor compression and variability structure descriptions. Moreover, while multilinear decomposition for tensors of 4th or higher order are occasionally used in Chemistry related fields, their applications in Social Sciences is uncommon. This is mainly due to model complexity which makes these tools unfriendly for non-experts. For tensors of compositions the degree of complexity increases even more, thus, compositional adaptations of 4th order decompositions are completely absent. Given these considerations the aim of this work is to address two issues which cause the infre- quent use of these tools, namely, parameter estimation ambiguities and interpretability concerns. In order to reach this goal, an application on a compositional 4th order is considered and a double optimization procedure algorithm is proposed as an alternative to standard Alternating Least Squares (ALS).