  We consider (robust) inference in the context of a factor model for
tensor-valued sequences. We study the consistency of the estimated common
factors and loadings space when using estimators based on minimising quadratic
loss functions. Building on the observation that such loss functions are
adequate only if sufficiently many moments exist, we extend our results to the
case of heavy-tailed distributions by considering estimators based on
minimising the Huber loss function, which uses an $L_{1}$ -norm weight on
outliers. We show that such class of estimators is robust to the presence of
heavy tails, even when only the second moment of the data exists. We also
propose a modified version of the eigenvalue-ratio principle to estimate the
dimensions of the core tensor and show the consistency of the resultant
estimators without any condition on the relative rates of divergence of the
sample size and dimensions. Extensive numerical studies are conducted to show
the advantages of the proposed methods over the state-of-the-art ones
especially under the heavy-tailed cases. An import/export dataset of a variety
of commodities across multiple countries is analyzed to show the practical
usefulness of the proposed robust estimation procedure. An R package ``RTFA"
implementing the proposed methods is available on R CRAN.
