We study the problem of dynamic assortment personalization with large, heterogeneous populations and wide arrays of products, and demonstrate the importance of structural priors for effective, efficient large-scale personalization. Assortment personalization is the problem of choosing, for each individual (type), a best assortment of products, ads, or other offerings (items) so as to maximize revenue. This problem is central to revenue management in e-commerce and online advertising where both items and types can number in the millions. We formulate the dynamic assortment personalization problem as a discrete-contextual bandit with $m$ contexts (types) and exponentially many arms (assortments of the $n$ items). We assume that each type's preferences follow a simple parametric model with $n$ parameters. In all, there are $mn$ parameters, and existing literature suggests that order optimal regret scales as $mn$. However, the data required to estimate so many parameters is orders of magnitude larger than the data available in most revenue management applications; and the optimal regret under these models is unacceptably high. In this paper, we impose a natural structure on the problem -- a small latent dimension, or low rank. In the static setting, we show that this model can be efficiently learned from surprisingly few interactions, using a time- and memory-efficient optimization algorithm that converges globally whenever the model is learnable. In the dynamic setting, we show that structure-aware dynamic assortment personalization can have regret that is an order of magnitude smaller than structure-ignorant approaches. We validate our theoretical results empirically.