Probabilistic phenomena in learning multi-index models in high dimension

A k-index model is a classical statistical model describing the dependency of a response variable y onto an input vector of covariates x. It posits that y depends on x only via its projection onto a k-dimensional subspace. Learning in this model boils down to estimating this subspace from data, a task that can be particularly challenging when x is high-dimensional. This setting encompasses a number of problems of practical interest, ranging from phase retrieval to small two-layer neural networks. I will give a condensed review of recent work on this model, emphasizing probability theory results and questions.