  Recommender systems have been widely adopted by electronic commerce and
entertainment industries for individualized prediction and recommendation,
which benefit consumers and improve business intelligence. In this article, we
propose an innovative method, namely the recommendation engine of multilayers
(REM), for tensor recommender systems. The proposed method utilizes the
structure of a tensor response to integrate information from multiple modes,
and creates an additional layer of nested latent factors to accommodate
between-subjects dependency. One major advantage is that the proposed method is
able to address the "cold-start" issue in the absence of information from new
customers, new products or new contexts. Specifically, it provides more
effective recommendations through sub-group information. To achieve scalable
computation, we develop a new algorithm for the proposed method, which
incorporates a maximum block improvement strategy into the cyclic
blockwise-coordinate-descent algorithm. In theory, we investigate both
algorithmic properties for global and local convergence, along with the
asymptotic consistency of estimated parameters. Finally, the proposed method is
applied in simulations and IRI marketing data with 116 million observations of
product sales. Numerical studies demonstrate that the proposed method
outperforms existing competitors in the literature.
