Survey on Algorithms for multi-index models

We review the literature on algorithms for estimating the in dex space in a multi-index model. The primary focus is on computa tionally efficient (polynomial-time) algorithms in Gaussian space, the assumptions under which consistency is guaranteed by these methods, and their sample complexity. In many cases, a gap is observed between the sample c omplexity of the best known computationally efficient methods and the i nformation-theoretical minimum. We also review algorithms based on est imating the span of gradients using nonparametric methods, and algorit hms based on fitting neural networks using gradient descent.