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 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.