Abstract: We study a class of graphical models having two layers of vertices, corresponding to latent and observable variables, all edges being directed from the latents to the observables. These and similar models were first studied in the neural networks literature, where they correspond to one level of computation; subsequently, such models have been studied in algebraic statistics. We are interested in the case that the model is invariant under permutation of the latent variables. We show: (a) When the latents are uniformly distributed, the model is identifiable at the least possible number of observables at which this might be so. (b) In the more general case of arbitrarily-distributed latents, the model is identifiable for a number of observables that is within a factor of two of best-possible. The only previous bound (to our knowledge) was exponentially larger. The proofs rely on root interlacing phenomena for some special three-term recurrences. Based on joint work with Spencer Gordon, Manav Kant, Eric Ma and Andrei Staicu.