Active seriation aims at recovering an unknown ordering of $n$ items by adaptively querying pairwise similarities. The observations are noisy measurements of entries of an underlying $n \times n$ permuted Robinson matrix, whose permutation encodes the latent ordering. The framework allows the algorithm to start with partial information on the latent ordering, including seriation from scratch as a special case. We propose an active seriation algorithm that provably recovers the latent ordering with high probability. Under a uniform separation condition on the similarity matrix, optimal performance guarantees are established, both in terms of the probability of error and the number of observations required for successful recovery.

We study the problem of estimating non-linear functionals of discrete distributions in the context of local differential privacy. The initial data $x_1,\ldots,x_n \in[K]$ are supposed i.i.d. and distributed according to an unknown discrete distribution $p = (p_1,\ldots,p_K)$. Only $\alpha$-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 $F_{\gamma} = \sum_{k=1}^K p_k^{\gamma}$, $\gamma >0$ as a function of $K, \, n$ and $\alpha$. In the non-interactive case, we study twol plug-in type estimators of $F_{\gamma}$, for all $\gamma >0$, that are similar to the MLE analyzed by Jiao et al. (2017) 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. (2020). In the sequentially interactive case, we introduce for all $\gamma >1$ a two-step procedure which attains the parametric rate $(n \alpha^2)^{-1/2}$ when $\gamma \geq 2$. We give lower bounds results over all $\alpha-$LDP mechanisms and over all estimators using the private samples.

Network complexity has been studied for over half a century and has found a wide range of applications. Many methods have been developed to characterize and estimate the complexity of networks. However, there has been little research with statistical guarantees. In this paper, we develop a statistical theory of graph complexity in a general model of random graphs, the so-called graphon model. Given a graphon, we endow the latent space of the nodes with the neighborhood distance. Our complexity index is then based on the covering number and the Minkowksi dimension of this metric space. Although the latent space is not identifiable, these indices turn out to be identifiable. This notion of complexity has simple interpretations on popular examples: it matches the number of communities in stochastic block models; the dimension of the Euclidean space in random geometric graphs; the regularity of the link function in Hölder graphons. From a single observation of the graph, we construct an estimator of the neighborhood-distance and show universal non-asymptotic bounds for its risk, matching minimax lower bounds. Based on this estimated distance, we compute the corresponding covering number and Minkowski dimension and we provide optimal non-asymptotic error bounds for these two plug-in estimators.