  In conventional statistical and machine learning methods, it is typically
assumed that the test data are identically distributed with the training data.
However, this assumption does not always hold, especially in applications where
the target population are not well-represented in the training data. This is a
notable issue in health-related studies, where specific ethnic populations may
be underrepresented, posing a significant challenge for researchers aiming to
make statistical inferences about these minority groups. In this work, we
present a novel approach to addressing this challenge in linear regression
models. We organize the model parameters for all the sub-populations into a
tensor. By studying a structured tensor completion problem, we can achieve
robust domain generalization, i.e., learning about sub-populations with limited
or no available data. Our method novelly leverages the structure of group
labels and it can produce more reliable and interpretable generalization
results. We establish rigorous theoretical guarantees for the proposed method
and demonstrate its minimax optimality. To validate the effectiveness of our
approach, we conduct extensive numerical experiments and a real data study
focused on education level prediction for multiple ethnic groups, comparing our
results with those obtained using other existing methods.
