  We describe a probabilistic PARAFAC/CANDECOMP (CP) factorization for multiway
(i.e., tensor) data that incorporates auxiliary covariates, SupCP. SupCP
generalizes the supervised singular value decomposition (SupSVD) for
vector-valued observations, to allow for observations that have the form of a
matrix or higher-order array. Such data are increasingly encountered in
biomedical research and other fields. We describe a likelihood-based latent
variable representation of the CP factorization, in which the latent variables
are informed by additional covariates. We give conditions for identifiability,
and develop an EM algorithm for simultaneous estimation of all model
parameters. SupCP can be used for dimension reduction, capturing latent
structures that are more accurate and interpretable due to covariate
supervision. Moreover, SupCP specifies a full probability distribution for a
multiway data observation with given covariate values, which can be used for
predictive modeling. We conduct comprehensive simulations to evaluate the SupCP
algorithm. We apply it to a facial image database with facial descriptors
(e.g., smiling / not smiling) as covariates, and to a study of amino acid
fluorescence. Software is available at https://github.com/lockEF/SupCP .
