arXiv subjects

Nicolas Verzelen

Publications and source records attributed to Nicolas Verzelen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Optimal rates for ranking a permuted isotonic matrix in polynomial time

We consider a ranking problem where we have noisy observations from a matrix with isotonic columns whose rows have been permuted by some permutation $π$ *. This encompasses many models, including crowd-labeling and ranking in tournaments by pair-wise comparisons. In this work, we provide an optimal and polynomial-time procedure for recovering $π$ * , settling an open problem in [7]. As a byproduct, our procedure is used to improve the state-of-the art for ranking problems in the stochastically transitive model (SST). Our approach is based on iterative pairwise comparisons by suitable data-driven weighted means of the columns. These weights are built using a combination of spectral methods with new dimension-reduction techniques. In order to deal with the important case of missing data, we establish a new concentration inequality for sparse and centered rectangular Wishart-type matrices.

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arXiv2023

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

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 $\sqrt{\log(n)/n}$, with high-probability. 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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arXiv2023

Active Ranking of Experts Based on their Performances in Many Tasks

We consider the problem of ranking n experts based on their performances on d tasks. We make a monotonicity assumption stating that for each pair of experts, one outperforms the other on all tasks. We consider the sequential setting where in each round, the learner has access to noisy evaluations of actively chosen pair of expert-task, given the information available up to the actual round. Given a confidence parameter $δ$ $\in$ (0, 1), we provide strategies allowing to recover the correct ranking of experts and develop a bound on the total number of queries made by our algorithm that hold with probability at least 1 -- $δ$. We show that our strategy is adaptive to the complexity of the problem (our bounds are instance dependent), and develop matching lower bounds up to a poly-logarithmic factor. Finally, we adapt our strategy to the relaxed problem of best expert identification and provide numerical simulation consistent with our theoretical results.

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arXiv2023

Optimal Permutation Estimation in Crowd-Sourcing problems

Motivated by crowd-sourcing applications, we consider a model where we have partial observations from a bivariate isotonic n x d matrix with an unknown permutation $π$ * acting on its rows. Focusing on the twin problems of recovering the permutation $π$ * and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way, we establish that, in some regimes, recovering the unknown permutation $π$ * is considerably simpler than estimating the matrix.

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arXiv2022

Optimal multiple change-point detection for high-dimensional data

This manuscript makes two contributions to the field of change-point detection. In a generalchange-point setting, we provide a generic algorithm for aggregating local homogeneity testsinto an estimator of change-points in a time series. Interestingly, we establish that the errorrates of the collection of tests directly translate into detection properties of the change-pointestimator. This generic scheme is then applied to various problems including covariance change-point detection, nonparametric change-point detection and sparse multivariate mean change-point detection. For the latter, we derive minimax optimal rates that are adaptive to theunknown sparsity and to the distance between change-points when the noise is Gaussian. Forsub-Gaussian noise, we introduce a variant that is optimal in almost all sparsity regimes.

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arXiv2021

Optimal Estimation of Schatten Norms of a rectangular Matrix

We consider the twin problems of estimating the effective rank and the Schatten norms $\|{\bf A}\|_{s}$ of a rectangular $p\times q$ matrix ${\bf A}$ from noisy observations. When $s$ is an even integer, we introduce a polynomial-time estimator of $\|{\bf A}\|_s$ that achieves the minimax rate $(pq)^{1/4}$. Interestingly, this optimal rate does not depend on the underlying rank of the matrix. When $s$ is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves the rate $(q\wedge p)(pq)^{1/4}$, which turns out to be optimal up to a logarithmic multiplicative term. The tight minimax rate is achieved by a more involved polynomial approximation method. This allows us to build estimators for a class of effective rank indices. As a byproduct, we also characterize the minimax rate for estimating the sequence of singular values of a matrix.

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arXiv2020

False discovery rate control with unknown null distribution: is it possible to mimic the oracle?

Classical multiple testing theory prescribes the null distribution, which is often a too stringent assumption for nowadays large scale experiments. This paper presents theoretical foundations to understand the limitations caused by ignoring the null distribution, and how it can be properly learned from the (same) data-set, when possible. We explore this issue in the case where the null distributions are Gaussian with an unknown rescaling parameters (mean and variance) and the alternative distribution is let arbitrary. While an oracle procedure in that case is the Benjamini Hochberg procedure applied with the true (unknown) null distribution, we pursue the aim of building a procedure that asymptotically mimics the performance of the oracle (AMO in short). Our main result states that an AMO procedure exists if and only if the sparsity parameter $k$ (number of false nulls) is of order less than $n/\log(n)$, where $n$ is the total number of tests. Further sparsity boundaries are derived for general location models where the shape of the null distribution is not necessarily Gaussian. Given our impossibility results, we also pursue a weaker objective, which is to find a confidence region for the oracle. To this end, we develop a distribution-dependent confidence region for the null distribution. As practical by-products, this provides a goodness of fit test for the null distribution, as well as a visual method assessing the reliability of empirical null multiple testing methods. Our results are illustrated with numerical experiments and a companion vignette \cite{RVvignette2020}.

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arXiv2020

Optimal Change-Point Detection and Localization

Given a times series ${\bf Y}$ in $\mathbb{R}^n$, with a piece-wise contant mean and independent components, the twin problems of change-point detection and change-point localization respectively amount to detecting the existence of times where the mean varies and estimating the positions of those change-points. In this work, we tightly characterize optimal rates for both problems and uncover the phase transition phenomenon from a global testing problem to a local estimation problem. Introducing a suitable definition of the energy of a change-point, we first establish in the single change-point setting that the optimal detection threshold is $\sqrt{2\log\log(n)}$. When the energy is just above the detection threshold, then the problem of localizing the change-point becomes purely parametric: it only depends on the difference in means and not on the position of the change-point anymore. Interestingly, for most change-point positions, it is possible to detect and localize them at a much smaller energy level. In the multiple change-point setting, we establish the energy detection threshold and show similarly that the optimal localization error of a specific change-point becomes purely parametric. Along the way, tight optimal rates for Hausdorff and $l_1$ estimation losses of the vector of all change-points positions are also established. Two procedures achieving these optimal rates are introduced. The first one is a least-squares estimator with a new multiscale penalty that favours well spread change-points. The second one is a two-step multiscale post-processing procedure whose computational complexity can be as low as $O(n\log(n))$. Notably, these two procedures accommodate with the presence of possibly many low-energy and therefore undetectable change-points and are still able to detect and localize high-energy change-points even with the presence of those nuisance parameters.

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arXiv2020

Detecting a planted community in an inhomogeneous random graph

We study the problem of detecting whether an inhomogeneous random graph contains a planted community. Specifically, we observe a single realization of a graph. Under the null hypothesis, this graph is a sample from an inhomogeneous random graph, whereas under the alternative, there exists a small subgraph where the edge probabilities are increased by a multiplicative scaling factor. We present a scan test that is able to detect the presence of such a planted community, even when this community is very small and the underlying graph is inhomogeneous. We also derive an information theoretic lower bound for this problem which shows that in some regimes the scan test is almost asymptotically optimal. We illustrate our results through examples and numerical experiments.

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arXiv2020

On the Estimation of Latent Distances Using Graph Distances

We are given the adjacency matrix of a geometric graph and the task of recovering the latent positions. We study one of the most popular approaches which consists in using the graph distances and derive error bounds under various assumptions on the link function. In the simplest case where the link function is proportional to an indicator function, the bound matches an information lower bound that we derive.

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arXiv2020

Optimal Sparsity Testing in Linear regression Model

We consider the problem of sparsity testing in the high-dimensional linear regression model. The problem is to test whether the number of non-zero components (aka the sparsity) of the regression parameter $θ^*$ is less than or equal to $k_0$. We pinpoint the minimax separation distances for this problem, which amounts to quantifying how far a $k_1$-sparse vector $θ^*$ has to be from the set of $k_0$-sparse vectors so that a test is able to reject the null hypothesis with high probability. Two scenarios are considered. In the independent scenario, the covariates are i.i.d. normally distributed and the noise level is known. In the general scenario, both the covariance matrix of the covariates and the noise level are unknown. Although the minimax separation distances differ in these two scenarios, both of them actually depend on $k_0$ and $k_1$ illustrating that for this composite-composite testing problem both the size of the null and of the alternative hypotheses play a key role.

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arXiv2020

Detection of Sparse Positive Dependence

In a bivariate setting, we consider the problem of detecting a sparse contamination or mixture component, where the effect manifests itself as a positive dependence between the variables, which are otherwise independent in the main component. We first look at this problem in the context of a normal mixture model. In essence, the situation reduces to a univariate setting where the effect is a decrease in variance. In particular, a higher criticism test based on the pairwise differences is shown to achieve the detection boundary defined by the (oracle) likelihood ratio test. We then turn to a Gaussian copula model where the marginal distributions are unknown. Standard invariance considerations lead us to consider rank tests. In fact, a higher criticism test based on the pairwise rank differences achieves the detection boundary in the normal mixture model, although not in the very sparse regime. We do not know of any rank test that has any power in that regime.

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arXiv2019

Partial recovery bounds for clustering with the relaxed $K$means

We investigate the clustering performances of the relaxed $K$means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decay exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed $K$means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed $K$means SDP allows to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed $K$means.

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arXiv2018

Model Assisted Variable Clustering: Minimax-optimal Recovery and Algorithms

Model-based clustering defines population level clusters relative to a model that embeds notions of similarity. Algorithms tailored to such models yield estimated clusters with a clear statistical interpretation. We take this view here and introduce the class of G-block covariance models as a background model for variable clustering. In such models, two variables in a cluster are deemed similar if they have similar associations will all other variables. This can arise, for instance, when groups of variables are noise corrupted versions of the same latent factor. We quantify the difficulty of clustering data generated from a G-block covariance model in terms of cluster proximity, measured with respect to two related, but different, cluster separation metrics. We derive minimax cluster separation thresholds, which are the metric values below which no algorithm can recover the model-defined clusters exactly, and show that they are different for the two metrics. We therefore develop two algorithms, COD and PECOK, tailored to G-block covariance models, and study their minimax-optimality with respect to each metric. Of independent interest is the fact that the analysis of the PECOK algorithm, which is based on a corrected convex relaxation of the popular K-means algorithm, provides the first statistical analysis of such algorithms for variable clustering. Additionally, we contrast our methods with another popular clustering method, spectral clustering, specialized to variable clustering, and show that ensuring exact cluster recovery via this method requires clusters to have a higher separation, relative to the minimax threshold. Extensive simulation studies, as well as our data analyses, confirm the applicability of our approach.

stat.ME↗

arXiv2018

Optimal graphon estimation in cut distance

Consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a W-random graph. We establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons. Surprisingly, our results imply that, from the minimax point of view, the raw data, that is, the adjacency matrix of the observed graph, is already optimal and more involved procedures cannot improve the convergence rates for this metric. This phenomenon contrasts with optimal rates of convergence with respect to other classical distances for graphons such as the l 1 or l 2 metrics.

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arXiv2018

Estimating minimum effect with outlier selection

We introduce one-sided versions of Huber's contamination model, in which corrupted samples tend to take larger values than uncorrupted ones. Two intertwined problems are addressed: estimation of the mean of uncorrupted samples (minimum effect) and selection of corrupted samples (outliers). Regarding the minimum effect estimation, we derive the minimax risks and introduce adaptive estimators to the unknown number of contaminations. Interestingly, the optimal convergence rate highly differs from that in classical Huber's contamination model. Also, our analysis uncovers the effect of particular structural assumptions on the distribution of the contaminated samples. As for the problem of selecting the outliers, we formulate the problem in a multiple testing framework for which the location/scaling of the null hypotheses are unknown. We rigorously prove how estimating the null hypothesis is possible while maintaining a theoretical guarantee on the amount of the falsely selected outliers, both through false discovery rate (FDR) or post hoc bounds. As a by-product, we address a long-standing open issue on FDR control under equi-correlation, which reinforces the interest of removing dependency when making multiple testing.

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arXiv2017

Oracle inequalities for network models and sparse graphon estimation

Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. We consider the problem of statistical estimation of the matrix of connection probabilities based on the observations of the adjacency matrix of the network. Taking the stochastic block model as an approximation, we construct estimators of network connection probabilities -- the ordinary block constant least squares estimator, and its restricted version. We show that they satisfy oracle inequalities with respect to the block constant oracle. As a consequence, we derive optimal rates of estimation of the probability matrix. Our results cover the important setting of sparse networks. Another consequence consists in establishing upper bounds on the minimax risks for graphon estimation in the $L\_2$ norm when the probability matrix is sampled according to a graphon model. These bounds include an additional term accounting for the "agnostic" error induced by the variability of the latent unobserved variables of the graphon model. In this setting, the optimal rates are influenced not only by the bias and variance components as in usual nonparametric problems but also include the third component, which is the agnostic error. The results shed light on the differences between estimation under the empirical loss (the probability matrix estimation) and under the integrated loss (the graphon estimation).

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arXiv2017

Adaptive estimation of High-Dimensional Signal-to-Noise Ratios

We consider the equivalent problems of estimating the residual variance, the proportion of explained variance $η$ and the signal strength in a high-dimensional linear regression model with Gaussian random design. Our aim is to understand the impact of not knowing the sparsity of the regression parameter and not knowing the distribution of the design on minimax estimation rates of $η$. Depending on the sparsity $k$ of the regression parameter, optimal estimators of $η$ either rely on estimating the regression parameter or are based on U-type statistics, and have minimax rates depending on $k$. In the important situation where $k$ is unknown, we build an adaptive procedure whose convergence rate simultaneously achieves the minimax risk over all $k$ up to a logarithmic loss which we prove to be non avoidable. Finally, the knowledge of the design distribution is shown to play a critical role. When the distribution of the design is unknown, consistent estimation of explained variance is indeed possible in much narrower regimes than for known design distribution.

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