Ery Arias-Castro

University of California, San Diego

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Nicolas Verzélen

Mathématiques, Informatique et Statistique pour l'Environnement et l'Agronomie

Bruno Pelletier

Centre National de la Recherche Scientifique

Wanli Qiao

Changchun University of Technology

David L. Donoho

Stanford University

Alberto Rodríguez‐Casal

Universidade de Santiago de Compostela

Clément Berenfeld

University of Potsdam

Gábor Lugosi

Barcelona School of Economics

Xiaoming Huo

Georgia Institute of Technology

Joseph Salmon

Université de Montpellier

Beatriz Pateiro‐López

Universidade de Santiago de Compostela

Cooperative Institutions

University of California, San Diego

Stanford University

Centre National de la Recherche Scientifique

Tongji University

Georgia Institute of Technology

UC San Diego Health System

Scripps Institution of Oceanography

University of California, Berkeley

California Institute of Technology

Duke University

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Distribution-Free Detection of Structured Anomalies: Permutation and Rank-Based Scans

Journal of the American Statistical Association (2017)

Ery Arias-Castro Rui M. Castro Ervin Tánczos Meng Wang

The scan statistic is by far the most popular method for anomaly detection, being popular in syndromic surveillance, signal and image processing, and target detection based on sensor networks, among other applications. The use of the scan statistics in such settings yields a hypothesis testing procedure, where the null hypothesis corresponds to the absence of anomalous behavior. If the null distribution is known, then calibration of a scan-based test is relatively easy, as it can be done by Monte Carlo simulation. When the null distribution is unknown, it is less straightforward. We investigate two procedures. The first one is a calibration by permutation and the other is a rank-based scan test, which is distribution-free and less sensitive to outliers. Furthermore, the rank scan test requires only a one-time calibration for a given data size making it computationally much more appealing. In both cases, we quantify the performance loss with respect to an oracle scan test that knows the null distribution. We show that using one of these calibration procedures results in only a very small loss of power in the context of a natural exponential family. This includes the classical normal location model, popular in signal processing, and the Poisson model, popular in syndromic surveillance. We perform numerical experiments on simulated data further supporting our theory and also on a real dataset from genomics. Supplementary materials for this article are available online.

Null (SQL)

Scan statistic

Rank (graph theory)

Resampling

10.1080/01621459.2017.1286240

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Community Detection in Sparse Random Networks

arXiv (Cornell University) (2014)

Nicolas Verzélen Ery Arias-Castro

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 Erdős–Renyi 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 $p_{1}>p_{0}$. In Arias-Castro and Verzelen [Ann. Statist. 42 (2014) 940–969], we focused on the asymptotically dense regime where $p_{0}$ is large enough that $np_{0}>(n/N)^{o(1)}$. We consider here the asymptotically sparse regime where $p_{0}$ is small enough that $np_{0} 0$. As before, we derive information theoretic lower bounds, and also establish the performance of various tests. Compared to our previous work [Ann. Statist. 42 (2014) 940–969], 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 tests statistics such as the size of the largest connected component, the number of triangles, and the number of subtrees of a given size. 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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Template Matching and Change Point Detection by M-Estimation

IEEE Transactions on Information Theory (2021)

Ery Arias-Castro Zheng Yan Lin

We consider the fundamental problem of matching a template to a signal. We do so by M-estimation, which encompasses procedures that are robust to gross errors (i.e., outliers). Using standard results from empirical process theory, we derive the convergence rate and the asymptotic distribution of the M-estimator under relatively mild assumptions. We also discuss the optimality of the estimator, both in finite samples in the minimax sense and in the large-sample limit in terms of local minimaxity and relative efficiency. Although most of the paper is dedicated to the study of the basic shift model in the context of a random design, we consider many extensions towards the end of the paper, including more flexible templates, fixed designs, the agnostic setting, and more.

Empirical distribution function

10.1109/tit.2021.3112680

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Estimation and variable selection with exponential weights

Electronic Journal of Statistics (2014)

Ery Arias-Castro Karim Lounici

In the context of a linear model with a sparse coefficient vector, exponential weights methods have been shown to be achieve oracle inequalities for denoising/prediction. We show that such methods also succeed at variable selection and estimation under the near minimum condition on the design matrix, instead of much stronger assumptions required by other methods such as the Lasso or the Dantzig Selector. The same analysis yields consistency results for Bayesian methods and BIC-type variable selection under similar conditions.

Lasso

Design matrix

10.1214/14-ejs883

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The Normalized Graph Cut and Cheeger Constant: From Discrete to Continuous

Advances in Applied Probability (2012)

Ery Arias-Castro Bruno Pelletier Pierre Pudlo

Let M be a bounded domain of with a smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M . By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M .

Normalization

Constant (computer programming)

10.1239/aap/1354716583

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Citations (31)

From Graph Centrality to Data Depth

HAL (Le Centre pour la Communication Scientifique Directe) (2021)

Eddie Aamari Ery Arias-Castro Clément Berenfeld

Given a sample of points in a Euclidean space, we can define a notion of depth by forming a neighborhood graph and applying a notion of centrality. In the present paper, we focus on the degree, iterates of the H-index, and the coreness, which are all well-known measures of centrality. We study their behaviors when applied to a sample of points drawn i.i.d. from an underlying density and with a connectivity radius properly chosen. Equivalently, we study these notions of centrality in the context of random neighborhood graphs. We show that, in the large-sample limit and under some standard condition on the connectivity radius, the degree converges to the likelihood depth (unsurprisingly), while iterates of the H-index and the coreness converge to new notions of depth.

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Cambridge University Press eBooks (2022)

Ery Arias-Castro

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

10.1017/9781108779197.029

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Oracle inequalities and minimax rates for non-local means and related adaptive kernel-based methods

arXiv (Cornell University) (2011)

Ery Arias-Castro Joseph Salmon Rebecca Willett

This paper describes a novel theoretical characterization of the performance of non-local means (NLM) for noise removal. NLM has proven effective in a variety of empirical studies, but little is understood fundamentally about how it performs relative to classical methods based on wavelets or how various parameters (e.g., patch size) should be chosen. For cartoon images and images which may contain thin features and regular textures, the error decay rates of NLM are derived and compared with those of linear filtering, oracle estimators, variable-bandwidth kernel methods, Yaroslavsky's filter and wavelet thresholding estimators. The trade-off between global and local search for matching patches is examined, and the bias reduction associated with the local polynomial regression version of NLM is analyzed. The theoretical results are validated via simulations for 2D images corrupted by additive white Gaussian noise.

Kernel (algebra)

10.48550/arxiv.1112.4434

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Finite Size Percolation in Regular Trees

arXiv (Cornell University) (2010)

Ery Arias-Castro

In the context of percolation in a regular tree, we study the size of the largest cluster and the length of the longest run starting within the first d generations. As d tends to infinity, we prove almost sure and weak convergence results.

Percolation (cognitive psychology)

Tree (set theory)

Infinity

10.48550/arxiv.1007.4508

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On the Consistency of the Crossmatch Test

arXiv (Cornell University) (2015)

Ery Arias-Castro Bruno Pelletier

Rosenbaum (2005) proposed the crossmatch test for two-sample goodness-of-fit testing in arbitrary dimensions. We prove that the test is consistent against all fixed alternatives. In the process, we develop a general consistency result based on (Henze & Penrose, 1999) that applies more generally.

Goodness of fit

Sample (material)

10.48550/arxiv.1509.05790

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