Supervised multiway factorization

Eric F. Lock, Gen Li

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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 use a novel 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

Original languageEnglish (US)
Pages (from-to)1150-1180
Number of pages31
JournalElectronic Journal of Statistics
Issue number1
StatePublished - 2018

Bibliographical note

Funding Information:
This work was supported in part by National Institutes of Health grant ULI RR033183/KL2 RR0333182 (to EFL).

Funding Information:
This work was supported in part by National Institutes of HealthNIH grant ULI RR033183/KL2 RR0333182 (to EFL).

Publisher Copyright:
© 2018 Informa UK Limited, trading as Taylor & Francis Group.


  • Dimension reduction
  • Faces in the wild
  • Latent variables
  • Parafac/candecomp
  • Singular value decomposition
  • Tensors


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