Optimal embedding on the sphere in non-parametric latent space models
Optimal embedding on the sphere in non-parametric latent space models
We consider the problem of embedding a set of items in a one-dimensional torus, using noisy observations of pairwise affinities. Importantly, the affinity function between items is unknown and it is only assumed that items with high affinity should be close in the latent space and that the affinity is a smooth function of latent positions. We introduce a new embedding procedure that provably uniformly localizes the latent positions of all items up to a precision of order $\sqrt{\log(n)/n}$. Conversely, this rate is proved to be minimax optimal.