  This paper studies zero-shot domain adaptation where each domain is indexed
on a multi-dimensional array, and we only have data from a small subset of
domains. Our goal is to produce predictors that perform well on \emph{unseen}
domains. We propose a model which consists of a domain-invariant latent
representation layer and a domain-specific linear prediction layer with a
low-rank tensor structure. Theoretically, we present explicit sample complexity
bounds to characterize the prediction error on unseen domains in terms of the
number of domains with training data and the number of data per domain. To our
knowledge, this is the first finite-sample guarantee for zero-shot domain
adaptation. In addition, we provide experiments on two-way MNIST and four-way
fiber sensing datasets to demonstrate the effectiveness of our proposed model.
