  We consider the problem of learning the preferences of a heterogeneous
population by observing choices from an assortment of products, ads, or other
offerings. Our observation model takes a form common in assortment planning
applications: each arriving customer is offered an assortment consisting of a
subset of all possible offerings; we observe only the assortment and the
customer's single choice.
  In this paper we propose a mixture choice model with a natural underlying
low-dimensional structure, and show how to estimate its parameters. In our
model, the preferences of each customer or segment follow a separate parametric
choice model, but the underlying structure of these parameters over all the
models has low dimension. We show that a nuclear-norm regularized maximum
likelihood estimator can learn the preferences of all customers using a number
of observations much smaller than the number of item-customer combinations.
This result shows the potential for structural assumptions to speed up learning
and improve revenues in assortment planning and customization. We provide a
specialized factored gradient descent algorithm and study the success of the
approach empirically.
