  As e-commerce expands, delivering real-time personalized recommendations from
vast catalogs poses a critical challenge for retail platforms. Maximizing
revenue requires careful consideration of both individual customer
characteristics and available item features to optimize assortments over time.
In this paper, we consider the dynamic assortment problem with dual contexts --
user and item features. In high-dimensional scenarios, the quadratic growth of
dimensions complicates computation and estimation. To tackle this challenge, we
introduce a new low-rank dynamic assortment model to transform this problem
into a manageable scale. Then we propose an efficient algorithm that estimates
the intrinsic subspaces and utilizes the upper confidence bound approach to
address the exploration-exploitation trade-off in online decision making.
Theoretically, we establish a regret bound of $\tilde{O}((d_1+d_2)r\sqrt{T})$,
where $d_1, d_2$ represent the dimensions of the user and item features
respectively, $r$ is the rank of the parameter matrix, and $T$ denotes the time
horizon. This bound represents a substantial improvement over prior literature,
made possible by leveraging the low-rank structure. Extensive simulations and
an application to the Expedia hotel recommendation dataset further demonstrate
the advantages of our proposed method.
