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 function 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 highprobability. This rate is proven to be minimax optimal. A computationally efficient variant of the procedure is also analyzed under some more restrictive assumptions. Our general results can be instantiated to the problem of statistical seriation, leading to new bounds for the maximum error in the ordering.

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We study the problem of estimating non-linear functionals of discrete distributions in the context of local differential privacy. The initial data x 1 , . . . , x n ∈ [K] are supposed i.i.d. and distributed according to an unknown discrete distribution p = (p 1 , . . . , p K ). Only α-locally differentially private (LDP) samples z 1 , ..., z n are publicly available, where the term 'local' means that each z i is produced using one individual attribute x i . We exhibit privacy mechanisms (PM) that are interactive (i.e. they are allowed to use already published confidential data) or non-interactive. We describe the behavior of the quadratic risk for estimating the power sum functional F γ = K k=1 p γ k , γ > 0 as a function of K, n and α. In the non-interactive case, we study two plug-in type estimators of F γ , for all γ > 0, that are similar to the MLE analyzed by Jiao et al. [14] in the multinomial model. However, due to the privacy constraint the rates we attain are slower and similar to those obtained in the Gaussian model by Collier et al. [6]. In the interactive case, we introduce for all γ > 1 a two-step procedure which attains the faster parametric rate (nα 2 ) −1/2 when γ ≥ 2. We give lower bounds results over all α−LDP mechanisms and all estimators using the private samples.