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We consider the problem of estimating latent positions in a one-dimensionaltorus from pairwise affinities. The observed affinity between a pair of itemsis modeled as a noisy observation of a function $f(x^*_{i},x^*_{j})$ of thelatent positions $x^*_{i},x^*_{j}$ of the two items on the torus. The affinityfunction $f$ is unknown, and it is only assumed to fulfill some shapeconstraints 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 goodflexibility to fit data. We introduce an estimation procedure that provablylocalizes all the latent positions with a maximum error of the order of$\sqrt{\log(n)/n}$, with high-probability. This rate is proven to be minimaxoptimal. A computationally efficient variant of the procedure is also analyzedunder some more restrictive assumptions. Our general results can beinstantiated to the problem of statistical seriation, leading to new bounds forthe maximum error in the ordering.

Localization in 1D non-parametric latent space models from pairwise affinities