  We introduce a tensor-based model of shared representation for meta-learning
from a diverse set of tasks. Prior works on learning linear representations for
meta-learning assume that there is a common shared representation across
different tasks, and do not consider the additional task-specific observable
side information. In this work, we model the meta-parameter through an
order-$3$ tensor, which can adapt to the observed task features of the task. We
propose two methods to estimate the underlying tensor. The first method solves
a tensor regression problem and works under natural assumptions on the data
generating process. The second method uses the method of moments under
additional distributional assumptions and has an improved sample complexity in
terms of the number of tasks.
  We also focus on the meta-test phase, and consider estimating task-specific
parameters on a new task. Substituting the estimated tensor from the first step
allows us estimating the task-specific parameters with very few samples of the
new task, thereby showing the benefits of learning tensor representations for
meta-learning. Finally, through simulation and several real-world datasets, we
evaluate our methods and show that it improves over previous linear models of
shared representations for meta-learning.
