Estimation of graph parameters based on a collection of graphs is essential for a wide range of graph inference tasks. In practice, weighted graphs are generally observed with edge contamination. We consider a weighted latent position graph model contaminated via an edge weight gross error model and propose an estimation methodology based on robust Lq estimation followed by low-rank adjacency spectral decomposition. We demonstrate that, under appropriate conditions, our estimator both maintains Lq robustness and wins the bias-variance tradeoff by exploiting low-rank graph structure. We illustrate the improvement offered by our estimator via both simulations and a human connectome data experiment.
There is a great need for robust techniques in data mining and machine learning contexts where many standard techniques such as principal component analysis and linear discriminant analysis are inherently susceptible to outliers. Furthermore, standard robust procedures assume that less than half the observation rows of a data matrix are contaminated, which may not be a realistic assumption when the number of observed features is large. This work looks at the problem of estimating covariance and precision matrices under cellwise contamination. We consider using a robust pairwise covariance matr…
Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The proposed estimator is shown to achieve minimax optimal rate under Huber's $ε$-contamination model for estimating covariance/sc…
Robust methods have been a successful approach to deal with contaminations and noises in image processing. In this paper, we introduce a new robust method for two-dimensional autoregressive models. Our method, called BMM-2D, relies on representing a two-dimensional autoregressive process with an auxiliary model to attenuate the effect of contamination (outliers). We compare the performance of our method with existing robust estimators and the least squares estimator via a comprehensive Monte Carlo simulation study which considers different levels of replacement contamination and window sizes. …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Cambridge（剑桥大学）
Robust estimation is an important problem in statistics which aims at providing a reasonable estimator when the data-generating distribution lies within an appropriately defined ball around an uncontaminated distribution. Although minimax rates of estimation have been established in recent years, many existing robust estimators with provably optimal convergence rates are also computationally intractable. In this paper, we study several estimation problems under a Wasserstein contamination model and present computationally tractable estimators motivated by generative adversarial networks (GANs)…
In data analysis, contamination caused by outliers is inevitable, and robust statistical methods are strongly demanded. In this paper, our concern is to develop a new approach for robust data analysis based on scoring rules. The scoring rule is a discrepancy measure to assess the quality of probabilistic forecasts. We propose a simple way of estimating not only the parameter in the statistical model but also the contamination ratio of outliers. Estimating the contamination ratio is important, since one can detect outliers out of the training samples based on the estimated contamination ratio. …
In the classical contamination models, such as the gross-error (Huber and Tukey contamination model or Case-wise Contamination), observations are considered as the units to be identified as outliers or not. This model is very useful when the number of considered variables is moderately small. Alqallaf et al. [2009] shows the limits of this approach for a larger number of variables and introduced the Independent contamination model (Cell-wise Contamination) where now the cells are the units to be identified as outliers or not. One approach to deal, at the same time, with both type of contaminat…
Multivariate location and scatter matrix estimation is a cornerstone in multivariate data analysis. We consider this problem when the data may contain independent cellwise and casewise outliers. Flat data sets with a large number of variables and a relatively small number of cases are common place in modern statistical applications. In these cases global down-weighting of an entire case, as performed by traditional robust procedures, may lead to poor results. We highlight the need for a new generation of robust estimators that can efficiently deal with cellwise outliers and at the same time sh…
Today's data pose unprecedented challenges to statisticians. It may be incomplete, corrupted or exposed to some unknown source of contamination. We need new methods and theories to grapple with these challenges. Robust estimation is one of the revived fields with potential to accommodate such complexity and glean useful information from modern datasets. Following our recent work on high dimensional robust covariance matrix estimation, we establish a general decision theory for robust statistics under Huber's $ε$-contamination model. We propose a solution using Scheff{é} estimate to a robust tw…
It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. In this paper, we start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further stud…
Robust Regression with Adaptive Contamination in Response: Optimal Rates and Computational Barriers
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Chicago（芝加哥大学）；University of California, San Diego（加州大学圣地亚哥分校）；Carnegie Mellon University（卡内基梅隆大学）
In this paper, we consider a parametric density contamination model. We work with a sample of i.i.d. data with a common density, $f^\star =(1-λ^\star) ϕ+ λ^\star ϕ(.-μ^\star)$, where the shape $ϕ$ is assumed to be known. We establish the optimal rates of convergence for the estimation of the mixture parameters $(λ^\star,μ^\star)$. In particular, we prove that the classical parametric rate $1/\sqrt{n}$ cannot be reached when at least one of these parameters is allowed to tend to $0$ with $n$.
In this paper, we develop connections between two seemingly disparate, but central, models in robust statistics: Huber's epsilon-contamination model and the heavy-tailed noise model. We provide conditions under which this connection provides near-statistically-optimal estimators. Building on this connection, we provide a simple variant of recent computationally-efficient algorithms for mean estimation in Huber's model, which given our connection entails that the same efficient sample-pruning based estimators is simultaneously robust to heavy-tailed noise and Huber contamination. Furthermore, w…
Johns Hopkins University（约翰斯·霍普金斯大学）；Microsoft（微软）；University of Maryland（马里兰大学）
Given a pair of graphs $G_1$ and $G_2$ and a vertex set of interest in $G_1$, the vertex nomination (VN) problem seeks to find the corresponding vertices of interest in $G_2$ (if they exist) and produce a rank list of the vertices in $G_2$, with the corresponding vertices of interest in $G_2$ concentrating, ideally, at the top of the rank list. In this paper, we define and derive the analogue of Bayes optimality for VN with multiple vertices of interest, and we define the notion of maximal consistency classes in vertex nomination. This theory forms the foundation for a novel VN adversarial con…
We consider the non-parametric regression problem under Huber's $ε$-contamination model, in which an $ε$ fraction of observations are subject to arbitrary adversarial noise. We first show that a simple local binning median step can effectively remove the adversary noise and this median estimator is minimax optimal up to absolute constants over the Hölder function class with smoothness parameters smaller than or equal to 1. Furthermore, when the underlying function has higher smoothness, we show that using local binning median as pre-preprocessing step to remove the adversarial noise, then we c…
Cox proportional hazard regression model is a popular tool to analyze the relationship between a censored lifetime variable with other relevant factors. The semi-parametric Cox model is widely used to study different types of data arising from applied disciplines like medical science, biology, reliability studies and many more. A fully parametric version of the Cox regression model, if properly specified, can yield more efficient parameter estimates leading to better insight generation. However, the existing maximum likelihood approach of generating inference under the fully parametric Cox reg…
Based on suitable left-truncated or censored data, two flexible classes of $M$-estimations of Weibull tail coefficient are proposed with two additional parameters bounding the impact of extreme contamination. Asymptotic normality with $\sqrt {n}$-rate of convergence is obtained. Its robustness is discussed via its asymptotic relative efficiency and influence function. It is further demonstrated by a small scale of simulations and an empirical study on CRIX.
Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-ε$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $α$-locally differentially private mechanism. The remaining $εn $ batches are an adver…
The setting of a right-censored random sample subject to contamination is considered. In various fields, expert information is often available and used to overcome the contamination. This paper integrates expert knowledge into the product-limit estimator in two different ways with distinct interpretations. Strong uniform consistency is proved for both cases under certain assumptions on the kind of contamination and the quality of expert information, which sheds light on the techniques and decisions that practitioners may take. The nuances of the techniques are discussed -- also with a view tow…
This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum $γ$-divergence estimator is well-known to work well estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribu…
Deviations from the center within a robust neighborhood of a parametric model distribution may naturally be considered an infinite dimensional nuisance parameter. Thus, the semiparametric method may be tried, which is to compute the scores function for the main parameter minus its orthogonal projection on the closed linear tangent space for the nuisance parameter, and then rescale for Fisher consistency. In this paper, we derive such a semiparametric influence curve by nonlinear projection on the tangent balls arising in robust statistics. This semiparametric influence curve is then compared w…
The parameters of the log-logistic distribution are generally estimated based on classical methods such as maximum likelihood estimation, whereas these methods usually result in severe biased estimates when the data contain outliers. In this paper, we consider several alternative estimators, which not only have closed-form expressions, but also are quite robust to a certain level of data contamination. We investigate the robustness property of each estimator in terms of the breakdown point. The finite sample performance and effectiveness of these estimators are evaluated through Monte Carlo si…
We introduce a robust and fully adaptive method for pointwise estimation in heteroscedastic regression. We allow for noise and design distributions that are unknown and fulfill very weak assumptions only. In particular, we do not impose moment conditions on the noise distribution. Moreover, we do not require a positive density for the design distribution. In a first step, we study the consistency of locally polynomial M-estimators that consist of a contrast and a kernel. Afterwards, minimax results are established over unidimensional Hölder spaces for degenerate design. We then choose the cont…
We provide a new computationally-efficient class of estimators for risk minimization. We show that these estimators are robust for general statistical models: in the classical Huber epsilon-contamination model and in heavy-tailed settings. Our workhorse is a novel robust variant of gradient descent, and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem. We provide specific consequences of our theory for linear regression, logistic regression and for estimation of the canonical parameters in an exponential f…
Robust estimation of a mean vector, a topic regarded as obsolete in the traditional robust statistics community, has recently surged in machine learning literature in the last decade. The latest focus is on the sub-Gaussian performance and computability of the estimators in a non-asymptotic setting. Numerous traditional robust estimators are computationally intractable, which partly contributes to the renewal of the interest in the robust mean estimation. Robust centrality estimators, however, include the trimmed mean and the sample median. The latter has the best robustness but suffers a low-…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We study the problem of outlier robust high-dimensional mean estimation under a finite covariance assumption, and more broadly under finite low-degree moment assumptions. We consider a standard stability condition from the recent robust statistics literature and prove that, except with exponentially small failure probability, there exists a large fraction of the inliers satisfying this condition. As a corollary, it follows that a number of recently developed algorithms for robust mean estimation, including iterative filtering and non-convex gradient descent, give optimal error estimators with …
This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_0)$ at some $x_0\in\mathbb{R}$ with i.i.d. observations, $$ X_1,\dots,X_n\sim (1-ε)f+εg, $$ where $g$ stands for a contamination distribution. In the context of multiple testing, this can be interpreted as estimating the null density at a point. We carefully study the effect of contamination on estimation through the following model indices: contamination proportion $ε$, smoothness of target density $β_0$, smoothness of contamination density $β_1$, and level of contami…
Graphical modeling explores dependences among a collection of variables by inferring a graph that encodes pairwise conditional independences. For jointly Gaussian variables, this translates into detecting the support of the precision matrix. Many modern applications feature high-dimensional and contaminated data that complicate this task. In particular, traditional robust methods that down-weight entire observation vectors are often inappropriate as high-dimensional data may feature partial contamination in many observations. We tackle this problem by giving a robust method for sparse precisio…
Google（谷歌）；University of California, Berkeley（加州大学伯克利分校）；Bell Labs（贝尔实验室）
This work is motivated by the problem of image mis-registration in remote sensing and we are interested in determining the resulting loss in the accuracy of pattern classification. A statistical formulation is given where we propose to use data contamination to model and understand the phenomenon of image mis-registration. This model is widely applicable to many other types of errors as well, for example, measurement errors and gross errors etc. The impact of data contamination on classification is studied under a statistical learning theoretical framework. A closed-form asymptotic bound is es…
The aim of the paper is to describe a model of the development of the Covid-19 contamination of the population of a country or a region. For this purpose a special branching process with two types of individuals is considered. This model is intended to use only the observed daily statistics to estimate the main parameter of the contamination and to give a prediction of the mean value of the non-observed population of the contaminated individuals. This is a serious advantage in comparison with other more complicated models where the observed official statistics are not sufficient. In this way t…
Consider longitudinal data $x_{ij},$ with $i=1,...,n$ and $j=1,...,p_{i},$ where $x_{ij}$ is the $j-$th observation of the random function $X_{i}\left( .\right) $ observed at time $t_{j}.$ The goal of this paper is to develop a parsimonious representation of the data by a linear combination of a set of $q$ smooth functions $H_{k}\left( .\right) $ ($k=1,..,q)$ in the sense that $x_{ij}\approxμ_{j}+\sum_{k=1}^{q}β_{ki}H_{k}\left( t_{j}\right) ,$ such that it fulfills three goals: it is resistant to atypical $X_{i}$'s ('case contamination'), it is resistant to isolated gross errors at some $t_{ij…
This paper describes a generalization of the Hellinger distance which we call the S -Hellinger distance; this general family connects the Hellinger distance smoothly with the $L_2$-divergence by a tuning parameter $α$ and is indeed a subfamily of the S -Divergence family of Ghosh et al. (2013 a, b). We use this general divergence in the context of estimating the location and covariances under (continuous) multivariate models and show that the proposed minimum S -Hellinger distance estimator is affine equivariant, asymptotically consistent and have high breakdown point under suitable conditions…
The dependency structure of multivariate data can be analyzed using the covariance matrix $Σ$. In many fields the precision matrix $Σ^{-1}$ is even more informative. As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. A popular high-dimensional estimator is the graphical lasso, but it lacks robustness. We consider the high-dimensional independent contamination model. Here, even a small percentage of contaminated cells in the data matrix may lead to a high percentage of contaminated rows. Downweighting entire observations,…
Classical semiparametric inference with missing outcome data is not robust to contamination of the observed data and a single observation can have arbitrarily large influence on estimation of a parameter of interest. This sensitivity is exacerbated when inverse probability weighting methods are used, which may overweight contaminated observations. We introduce inverse probability weighted, double robust and outcome regression estimators of location and scale parameters, which are robust to contamination in the sense that their influence function is bounded. We give asymptotic properties and st…
