  Single-index models are a class of functions given by an unknown univariate
``link'' function applied to an unknown one-dimensional projection of the
input. These models are particularly relevant in high dimension, when the data
might present low-dimensional structure that learning algorithms should adapt
to. While several statistical aspects of this model, such as the sample
complexity of recovering the relevant (one-dimensional) subspace, are
well-understood, they rely on tailored algorithms that exploit the specific
structure of the target function. In this work, we introduce a natural class of
shallow neural networks and study its ability to learn single-index models via
gradient flow. More precisely, we consider shallow networks in which biases of
the neurons are frozen at random initialization. We show that the corresponding
optimization landscape is benign, which in turn leads to generalization
guarantees that match the near-optimal sample complexity of dedicated
semi-parametric methods.
