We consider the problem of estimating latent positions in a one-dimensional torus from pairwise affinities. The observed affinity between a pair of items is modeled as a noisy observation of a function f (x*i , x*j) of the latent positions x*i, x*j of the two items on the torus. The affinity func-tion f is unknown, and it is only assumed to fulfill some shape constraints ensuring that f(x, y) is large when the distance between x and y is small, and vice-versa. This non-parametric modeling offers a good flexibility to fit data. We introduce an estimation procedure that provably localizes all the latent positions with a maximum error of the order of log(n)/n, with high-probability. This rate is proven to be minimax optimal. A computa-tionally efficient variant of the procedure is also analyzed under some more restrictive assumptions. Our general results can be instantiated to the prob-lem of statistical seriation, leading to new bounds for the maximum error in the ordering.