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Nicolas Verzelen

Publications and source records attributed to Nicolas Verzelen.

At least 55 records · Page 3Linked to original sources
arXiv2017

Adaptive estimation of the sparsity in the Gaussian vector model

Consider the Gaussian vector model with mean value θ. We study the twin problems of estimating the number |θ|_0 of non-zero components of θ and testing whether |θ|_0 is smaller than some value. For testing, we establish the minimax separation distances for this model and introduce a minimax adaptive test. Extensions to the case of unknown variance are also discussed. Rewriting the estimation of |θ|_0 as a multiple testing problem of all hypotheses {|θ|_0 <= q}, we both derive a new way of assessing the optimality of a sparsity estimator and we exhibit such an optimal procedure. This general approach provides a roadmap for estimating the complexity of the signal in various statistical models.

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arXiv2017

Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization

We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. Our bounds differ by at most a factor of root two when the rank is large (in the clustering and submatrix localization problems, when the number of clusters or blocks is large) or the signal matrix is very sparse. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information- theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs.

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arXiv2016

Detecting a Path of Correlations in a Network

We consider the problem of detecting an anomaly in the form of a path of correlations hidden in white noise. We provide a minimax lower bound and a test that, under mild assumptions, is able to achieve the lower bound up to a multiplicative constant.

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arXiv2016

Detection and Feature Selection in Sparse Mixture Models

We consider Gaussian mixture models in high dimensions and concentrate on the twin tasks of detection and feature selection. Under sparsity assumptions on the difference in means, we derive information bounds and establish the performance of various procedures, including the top sparse eigenvalue of the sample covariance matrix and other projection tests based on moments, such as the skewness and kurtosis tests of Malkovich and Afifi (1973), and other variants which we were better able to control under the null.

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arXiv2016

PECOK: a convex optimization approach to variable clustering

The problem of variable clustering is that of grouping similar components of a $p$-dimensional vector $X=(X_{1},\ldots,X_{p})$, and estimating these groups from $n$ independent copies of $X$. When cluster similarity is defined via $G$-latent models, in which groups of $X$-variables have a common latent generator, and groups are relative to a partition $G$ of the index set $\{1, \ldots, p\}$, the most natural clustering strategy is $K$-means. We explain why this strategy cannot lead to perfect cluster recovery and offer a correction, based on semi-definite programing, that can be viewed as a penalized convex relaxation of $K$-means (PECOK). We introduce a cluster separation measure tailored to $G$-latent models, and derive its minimax lower bound for perfect cluster recovery. The clusters estimated by PECOK are shown to recover $G$ at a near minimax optimal cluster separation rate, a result that holds true even if $K$, the number of clusters, is estimated adaptively from the data. We compare PECOK with appropriate corrections of spectral clustering-type procedures, and show that the former outperforms the latter for perfect cluster recovery of minimally separated clusters.

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arXiv2015

Detecting Markov Random Fields Hidden in White Noise

Motivated by change point problems in time series and the detection of textured objects in images, we consider the problem of detecting a piece of a Gaussian Markov random field hidden in white Gaussian noise. We derive minimax lower bounds and propose near-optimal tests.

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arXiv2014

Community Detection in Sparse Random Networks

We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph on $N$ vertices and with connection probability $p_0$; under the alternative, there is an unknown subgraph on $n$ vertices where the connection probability is p1 > p0. In Arias-Castro and Verzelen (2012), we focused on the asymptotically dense regime where p0 is large enough that np0>(n/N)^{o(1)}. We consider here the asymptotically sparse regime where p0 is small enough that np0<(n/N)^{c0} for some c0>0. As before, we derive information theoretic lower bounds, and also establish the performance of various tests. Compared to our previous work, the arguments for the lower bounds are based on the same technology, but are substantially more technical in the details; also, the methods we study are different: besides a variant of the scan statistic, we study other statistics such as the size of the largest connected component, the number of triangles, the eigengap of the adjacency matrix, etc. Our detection bounds are sharp, except in the Poisson regime where we were not able to fully characterize the constant arising in the bound.

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arXiv2014

A Global Homogeneity Test for High-Dimensional Linear Regression

This paper is motivated by the comparison of genetic networks based on microarray samples. The aim is to test whether the differences observed between two inferred Gaussian graphical models come from real differences or arise from estimation uncertainties. Adopting a neighborhood approach, we consider a two-sample linear regression model with random design and propose a procedure to test whether these two regressions are the same. Relying on multiple testing and variable selection strategies, we develop a testing procedure that applies to high-dimensional settings where the number of covariates $p$ is larger than the number of observations $n_1$ and $n_2$ of the two samples. Both type I and type II errors are explicitely controlled from a non-asymptotic perspective and the test is proved to be minimax adaptive to the sparsity. The performances of the test are evaluated on simulated data. Moreover, we illustrate how this procedure can be used to compare genetic networks on Hess \emph{et al} breast cancer microarray dataset.

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arXiv2013

Community Detection in Random Networks

We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph with probability p0. Under the (composite) alternative, there is a subgraph of n nodes where the probability of connection is p1 > p0. We derive a detection lower bound for detecting such a subgraph in terms of N, n, p0, p1 and exhibit a test that achieves that lower bound. We do this both when p0 is known and unknown. We also consider the problem of testing in polynomial-time. As an aside, we consider the problem of detecting a clique, which is intimately related to the planted clique problem. Our focus in this paper is in the quasi-normal regime where n p0 is either bounded away from zero, or tends to zero slowly.

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arXiv2013

Minimax adaptive tests for the Functional Linear model

We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional Principal Component Analysis. Interestingly, the procedures are completely data-driven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local alternatives are assessed in a nonasymptotic setting. This allows us to prove that these procedures are minimax adaptive (up to an unavoidable \log\log n multiplicative term) to the unknown regularity of the slope. As a side result, the minimax separation distances of the slope are derived for a large range of regularity classes. A numerical study illustrates these theoretical results.

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arXiv2012

High-dimensional regression with unknown variance

We review recent results for high-dimensional sparse linear regression in the practical case of unknown variance. Different sparsity settings are covered, including coordinate-sparsity, group-sparsity and variation-sparsity. The emphasis is put on non-asymptotic analyses and feasible procedures. In addition, a small numerical study compares the practical performance of three schemes for tuning the Lasso estimator and some references are collected for some more general models, including multivariate regression and nonparametric regression.

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arXiv2012

Graph selection with GGMselect

Applications on inference of biological networks have raised a strong interest in the problem of graph estimation in high-dimensional Gaussian graphical models. To handle this problem, we propose a two-stage procedure which first builds a family of candidate graphs from the data, and then selects one graph among this family according to a dedicated criterion. This estimation procedure is shown to be consistent in a high-dimensional setting, and its risk is controlled by a non-asymptotic oracle-like inequality. The procedure is tested on a real data set concerning gene expression data, and its performances are assessed on the basis of a large numerical study. The procedure is implemented in the R-package GGMselect available on the CRAN.

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arXiv2012

Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons

Consider the standard Gaussian linear regression model $Y=Xθ+ε$, where $Y\in R^n$ is a response vector and $ X\in R^{n*p}$ is a design matrix. Numerous work have been devoted to building efficient estimators of $θ$ when $p$ is much larger than $n$. In such a situation, a classical approach amounts to assume that $θ_0$ is approximately sparse. This paper studies the minimax risks of estimation and testing over classes of $k$-sparse vectors $θ$. These bounds shed light on the limitations due to high-dimensionality. The results encompass the problem of prediction (estimation of $Xθ$), the inverse problem (estimation of $θ_0$) and linear testing (testing $Xθ=0$). Interestingly, an elbow effect occurs when the number of variables $k\log(p/k)$ becomes large compared to $n$. Indeed, the minimax risks and hypothesis separation distances blow up in this ultra-high dimensional setting. We also prove that even dimension reduction techniques cannot provide satisfying results in an ultra-high dimensional setting. Moreover, we compute the minimax risks when the variance of the noise is unknown. The knowledge of this variance is shown to play a significant role in the optimal rates of estimation and testing. All these minimax bounds provide a characterization of statistical problems that are so difficult so that no procedure can provide satisfying results.

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arXiv2010

Adaptive estimation of covariance matrices via Cholesky decomposition

This paper studies the estimation of a large covariance matrix. We introduce a novel procedure called ChoSelect based on the Cholesky factor of the inverse covariance. This method uses a dimension reduction strategy by selecting the pattern of zero of the Cholesky factor. Alternatively, ChoSelect can be interpreted as a graph estimation procedure for directed Gaussian graphical models. Our approach is particularly relevant when the variables under study have a natural ordering (e.g. time series) or more generally when the Cholesky factor is approximately sparse. ChoSelect achieves non-asymptotic oracle inequalities with respect to the Kullback-Leibler entropy. Moreover, it satisfies various adaptive properties from a minimax point of view. We also introduce and study a two-stage procedure that combines ChoSelect with the Lasso. This last method enables the practitioner to choose his own trade-off between statistical efficiency and computational complexity. Moreover, it is consistent under weaker assumptions than the Lasso. The practical performances of the different procedures are assessed on numerical examples.

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arXiv2010

Detection boundary in sparse regression

We study the problem of detection of a p-dimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to the infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the high-dimensional setting (p>> n). The main message is that, under some conditions, the detection boundary phenomenon that has been proved for the Gaussian sequence model, extends to high-dimensional linear regression. Finally, we establish the detection boundaries when the variance of the noise is unknown. Interestingly, the detection boundaries sometimes depend on the knowledge of the variance in a high-dimensional setting.

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arXiv2009

Adaptive estimation of stationary Gaussian fields

We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a regular lattice. In the time series setting, some procedures like AIC are proved to achieve optimal model selection among autoregressive models. However, there exists no such equivalent results of adaptivity in a spatial setting. By considering collections of Gaussian Markov random fields (GMRF) as approximation sets for the distribution of X, we introduce a novel model selection procedure for spatial fields. For all neighborhoods m in a given collection M, this procedure first amounts to computing a covariance estimator of X within the GMRFs of neighborhood m. Then, it selects a neighborhood by applying a penalization strategy. The so-defined method satisfies a nonasymptotic oracle type inequality. If X is a GMRF, the procedure is also minimax adaptive to the sparsity of its neighborhood. More generally, the procedure is adaptive to the rate of approximation of the true distribution by GMRFs with growing neighborhoods.

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arXiv2009

Data-driven neighborhood selection of a Gaussian field

We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a lattice. To tackle this issue, a neighborhood selection procedure has been recently introduced. This procedure amounts to selecting a neighborhood m by a penalization method and estimating the covariance of X in the space of Gaussian Markov random fields (GMRFs) with neighborhood m. Such a strategy is shown to satisfy oracle inequalities as well as minimax adaptive properties. However, it suffers several drawbacks which make the method difficult to apply in practice. On the one hand, the penalty depends on some unknown quantities. On the other hand, the procedure is only defined for toroidal lattices. The present contribution is threefold. A data-driven algorithm is proposed for tuning the penalty function. Moreover, the procedure is extended to non-toroidal lattices. Finally, numerical study illustrate the performances of the method on simulated examples. These simulations suggest that Gaussian Markov random field selection is often a good alternative to variogram estimation.

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