Principal Component Analysis for Multivariate Extremes
This chapter explores ways to reduce the dimensionality of the data while preserving key information relevant to the analysis of multivariate extreme values.
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arXiv subjects
Anne Sabourin
Publications and source records attributed to Anne Sabourin.
Out-of-Distribution generalization of quantile regression with heavy tailed inputs: an SVM approach
We study quantile regression in an extrapolation regime where the covariate takes unusually large values. Under regular variation assumptions, extreme observations can be effectively characterized through their angular components, enabling learning strategies that focus on the angle of the most extreme observations. This approach is formalized through the minimization of an asymptotic conditional risk that localizes learning in the tail of the covariate distribution. We propose a novel Support Vector Machine (SVM) framework for extreme quantile regression, leveraging reproducing kernel Hilbert spaces to handle high-dimensional and nonlinear settings. Our method also accommodates unbounded response variables and avoids restrictive transformations. We establish finite-sample learning guarantees under mild regularity assumptions. The proposed framework unifies ideas from statistical learning and multivariate extremes, providing a tractable and theoretically grounded approach to extrapolation. We complement our theoretical findings with an empirical study on river flow data from the Danube, demonstrating the practical relevance of our methods.
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Polar Depth for Potentially Heavy-Tailed Data
Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as Polar Depth. The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a regularly varying random variable, beyond asymptotically large thresholds, once its marginals have been appropriately normalized. Not only does the polar depth function make it easy to order the extreme values taken by a heavy-tailed random variable X and finds natural applications in anomaly detection, but it is also possible to show, as we prove it under appropriate assumptions in this article, that the polar depth of the largest observations, i.e. observations X which norm is larger than t>0, converges to the polar depth of the limiting distribution as t converges to infinity. Although designed to quantify the depth of multivariate extremes, the polar depth is interesting in its own right, insofar as this notion is more relevant for distributions whose support is included in a halfspace than the alternatives proposed in the literature, the halfspace depth in particular. Here, we demonstrate its properties and analyze statistical issues related to its estimation from both finite-sample and asymptotic points of view. We present numerical results to empirically demonstrate its relevance, particularly for the statistical analysis of extreme observations and more specifically for the identification of anomalies among them.
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Zero-couplings of infinite measures with cyclically monotone support and multivariate regular variation
We study cyclically monotone transport plans between measures in $\mathrm{M}_0(\mathbb{R}^d)$, the class of Borel measures on $\mathbb{R}^d \setminus \{0\}$ that are finite on sets bounded away from the origin but may have infinite total mass. We avoid moment assumptions and allow the transport cost to be infinite. This framework naturally arises for exponent measures in multivariate regular variation and includes other examples such as L\'evy measures. We introduce the notion of a zero-coupling and establish existence of cyclically monotone zero-couplings for arbitrary pairs of measures in $\mathrm{M}_0(\mathbb{R}^d)$. Under a Hausdorff-dimension condition on the first measure and when at least one of the two measures has infinite mass, we prove uniqueness of the cyclically monotone zero-coupling, yielding an analogue of the Brenier--McCann theorem in this infinite-measure setting. We further derive a representation of such couplings through gradients of closed convex functions and identify conditions under which the zero-coupling is proper in the sense that the second measure is equal to the restriction to the punctured space of the push-forward of the first measure by a cyclically monotone transport map. Finally, we apply these results to regularly varying probability measures. We show that a cyclically monotone coupling between two such distributions admits a tail limit that coincides with the unique proper cyclically monotone zero-coupling between the corresponding exponent measures.
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Extrapolation in Statistical Learning with Extreme Value Theory
Extreme value theory provides rigorous theory and statistical tools for extrapolation in machine learning, particularly in settings where traditional methods struggle due to data scarcity in the tails. A broad range of tasks benefit from these advances, including regression and classification beyond the training data, extreme quantile regression, supervised and unsupervised dimension reduction, generative artificial intelligence and anomaly detection. This review synthesizes recent developments in these fields at the intersection of statistical learning and extreme value theory, with a focus on principled methods based on asymptotically motivated representations of the tail of univariate and multivariate distributions. We consider different theoretical frameworks for both asymptotically dependent and independent data and discuss how they translate into efficient statistical methods for extrapolation to extreme regions. By addressing both theoretical and practical aspects, we offer a comprehensive overview of the state-of-the-art in this quickly evolving field, and identify promising directions for future research.
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Adaptive tail index estimation: minimal assumptions and non-asymptotic guarantees
A notoriously difficult challenge in extreme value theory is the choice of the number $k\ll n$, where $n$ is the total sample size, of extreme data points to consider for inference of tail quantities. Existing theoretical guarantees for adaptive methods typically require second-order assumptions or von Mises assumptions that are difficult to verify and often come with tuning parameters that are challenging to calibrate. This paper revisits the problem of adaptive selection of $k$ for the Hill estimator. Our goal is not an `optimal' $k$ but one that is `good enough', in the sense that we strive for non-asymptotic guarantees that might be sub-optimal but are explicit and require minimal conditions. We propose a transparent adaptive rule that does not require preliminary calibration of constants, inspired by `adaptive validation' developed in high-dimensional statistics. A key feature of our approach is the consideration of a grid for $k$ of size $ \ll n $, which aligns with common practice among practitioners but has remained unexplored in theoretical analysis. Our rule only involves an explicit expression of a variance-type term; in particular, it does not require controlling or estimating a biasterm. Our theoretical analysis is valid for all heavy-tailed distributions, specifically for all regularly varying survival functions. Furthermore, when von Mises conditions hold, our method achieves `almost' minimax optimality with a rate of $\sqrt{\log \log n}~ n^{-|\rho|/(1+2|\rho|)}$ when the grid size is of order $\log n$, in contrast to the $ (\log \log (n)/n)^{|\rho|/(1+2|\rho|)} $ rate in existing work. Our simulations show that our approach performs particularly well for ill-behaved distributions.
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Multi-site modelling and reconstruction of past extreme skew surges along the French Atlantic coast
Appropriate modelling of extreme skew surges is crucial, particularly for coastal risk management. Our study focuses on modelling extreme skew surges along the French Atlantic coast, with a particular emphasis on investigating the extremal dependence structure between stations. We employ the peak-over-threshold framework, where a multivariate extreme event is defined whenever at least one location records a large value, though not necessarily all stations simultaneously. A novel method for determining an appropriate level (threshold) above which observations can be classified as extreme is proposed. Two complementary approaches are explored. First, the multivariate generalized Pareto distribution is employed to model extremes, leveraging its properties to derive a generative model that predicts extreme skew surges at one station based on observed extremes at nearby stations. Second, a novel extreme regression framework is assessed for point predictions. This specific regression framework enables accurate point predictions using only the 'angle' of input variables, i.e., input variables divided by their norms. The ultimate objective is to reconstruct historical skew surge time series at stations with limited data. This is achieved by integrating extreme skew surge data from stations with longer records, such as Brest and Saint-Nazaire, which provide over 150 years of observations.
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Weak Signals and Heavy Tails: Learning Theory meets Extreme Value Analysis
The masses of data now available have opened up the prospect of discovering weak signals using machine-learning algorithms, with a view to predictive or interpretation tasks. As this survey of recent results attempts to show, bringing multivariate extreme value theory and statistical learning theory together in a common, nonparametric and nonasymptotic framework makes it possible to design and analyze new methods for exploiting the scarce information located in distribution tails in these purposes. This article reviews recently proved theoretical tools for establishing guarantees for supervised or unsupervised algorithms learning from a fraction of extreme data. These are mainly exponential maximal deviation inequalities tailored to low-probability regions and concentration results for stochastic processes empirically describing the behavior of multivariate extreme observations, their dependence structure in particular. Under appropriate assumptions of regular variation, several illustrative applications in multivariate settings are then examined: classification, regression, anomaly detection, model selection via cross-validation. For these, generalization results are established inspired by the classical bounds in statistical learning theory. In the same spirit, it is also shown how to adapt the popular high-dimensional Lasso technique in the context of extreme values for the covariates with generalization guarantees.
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Sharp error bounds for imbalanced classification: how many examples in the minority class?
When dealing with imbalanced classification data, reweighting the loss function is a standard procedure allowing to equilibrate between the true positive and true negative rates within the risk measure. Despite significant theoretical work in this area, existing results do not adequately address a main challenge within the imbalanced classification framework, which is the negligible size of one class in relation to the full sample size and the need to rescale the risk function by a probability tending to zero. To address this gap, we present two novel contributions in the setting where the rare class probability approaches zero: (1) a non asymptotic fast rate probability bound for constrained balanced empirical risk minimization, and (2) a consistent upper bound for balanced nearest neighbors estimates. Our findings provide a clearer understanding of the benefits of class-weighting in realistic settings, opening new avenues for further research in this field.
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Regular Variation in Hilbert Spaces and Principal Component Analysis for Functional Extremes
Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal `directions' of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.
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On the bias of K-fold cross validation with stable learners
This paper investigates the efficiency of the K-fold cross-validation (CV) procedure and a debiased version thereof as a means of estimating the generalization risk of a learning algorithm. We work under the general assumption of uniform algorithmic stability. We show that the K-fold risk estimate may not be consistent under such general stability assumptions, by constructing non vanishing lower bounds on the error in realistic contexts such as regularized empirical risk minimisation and stochastic gradient descent. We thus advocate the use of a debiased version of the K-fold and prove an error bound with exponential tail decay regarding this version. Our result is applicable to the large class of uniformly stable algorithms, contrarily to earlier works focusing on specific tasks such as density estimation. We illustrate the relevance of the debiased K-fold CV on a simple model selection problem and demonstrate empirically the usefulness of the promoted approach on real world classification and regression datasets.
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On Regression in Extreme Regions
We establish a statistical learning theoretical framework aimed at extrapolation, or out-of-domain generalization, on the unobserved tails of covariates in continuous regression problems. Our strategy involves performing statistical regression on a subsample of observations with continuous labels that are the furthest away from the origin, focusing specifically on their angular components. The underlying assumptions of our approach are grounded in the theory of multivariate regular variation, a cornerstone of extreme value theory. We address the stylized problem of nonparametric least squares regression with predictors chosen from a Vapnik-Chervonenkis class. This work contributes to a broader initiative to develop statistical learning theoretical foundations for supervised learning strategies that enhance performance on the supposedly heavy tails of covariates. Previous efforts in this area have focused exclusively on binary classification on extreme covariates. Although the continuous target setting necessitates different techniques and regularity assumptions, our main results echo findings from earlier studies. We quantify the predictive performance on tail regions in terms of excess risk, presenting it as a finite sample risk bound with a clear bias-variance decomposition. Numerical experiments with simulated and real data illustrate our theoretical findings.
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Tail inverse regression for dimension reduction with extreme response
We consider the problem of supervised dimension reduction with a particular focus on extreme values of the target $Y\in\mathbb{R}$ to be explained by a covariate vector $X \in \mathbb{R}^p$. The general purpose is to define and estimate a projection on a lower dimensional subspace of the covariate space which is sufficient for predicting exceedances of the target above high thresholds. We propose an original definition of Tail Conditional Independence which matches this purpose. Inspired by Sliced Inverse Regression (SIR) methods, we develop a novel framework (TIREX, Tail Inverse Regression for EXtreme response) in order to estimate an extreme sufficient dimension reduction (SDR) space of potentially smaller dimension than that of a classical SDR space. We prove the weak convergence of tail empirical processes involved in the estimation procedure and we illustrate the relevance of the proposed approach on simulated and real world data.
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Concentration bounds for the empirical angular measure with statistical learning applications
The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. However, the study of the sampling distribution of the resulting empirical angular measure is challenging. It is the purpose of the paper to establish finite-sample bounds for the maximal deviations between the empirical and true angular measures, uniformly over classes of Borel sets of controlled combinatorial complexity. The bounds are valid with high probability and, up to logarithmic factors, scale as the square root of the effective sample size. The bounds are applied to provide performance guarantees for two statistical learning procedures tailored to extreme regions of the input space and built upon the empirical angular measure: binary classification in extreme regions through empirical risk minimization and unsupervised anomaly detection through minimum-volume sets of the sphere.
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Uniform concentration bounds for frequencies of rare events
New Vapnik and Chervonenkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on the maximal deviation over classes of Borel sets within a low probability region. The constants are explicit, enabling numerical comparisons.
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Cross-validation on Extreme Regions
We conduct a non asymptotic study of the Cross Validation (CV) estimate of the generalization risk for learning algorithms dedicated to extreme regions of the covariates space. In this Extreme Value Analysis context, the risk function measures the algorithm's error given that the norm of the input exceeds a high quantile. The main challenge within this framework is the negligible size of the extreme training sample with respect to the full sample size and the necessity to re-scale the risk function by a probability tending to zero. We open the road to a finite sample understanding of CV for extreme values by establishing two new results: an exponential probability bound on the \Kfold CV error and a polynomial probability bound on the leave-\textrm{p}-out CV. Our bounds are sharp in the sense that they match state-of-the-art guarantees for standard CV estimates while extending them to encompass a conditioning event of small probability. We illustrate the significance of our results regarding high dimensional classification in extreme regions via a Lasso-type logistic regression algorithm. The tightness of our bounds is investigated in numerical experiments.
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Heavy-tailed Representations, Text Polarity Classification & Data Augmentation
The dominant approaches to text representation in natural language rely on learning embeddings on massive corpora which have convenient properties such as compositionality and distance preservation. In this paper, we develop a novel method to learn a heavy-tailed embedding with desirable regularity properties regarding the distributional tails, which allows to analyze the points far away from the distribution bulk using the framework of multivariate extreme value theory. In particular, a classifier dedicated to the tails of the proposed embedding is obtained which performance outperforms the baseline. This classifier exhibits a scale invariance property which we leverage by introducing a novel text generation method for label preserving dataset augmentation. Numerical experiments on synthetic and real text data demonstrate the relevance of the proposed framework and confirm that this method generates meaningful sentences with controllable attribute, e.g. positive or negative sentiment.
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A Multivariate Extreme Value Theory Approach to Anomaly Clustering and Visualization
In a wide variety of situations, anomalies in the behaviour of a complex system, whose health is monitored through the observation of a random vector X = (X1,. .. , X d) valued in R d , correspond to the simultaneous occurrence of extreme values for certain subgroups $α$ $\subset$ {1,. .. , d} of variables Xj. Under the heavy-tail assumption, which is precisely appropriate for modeling these phenomena, statistical methods relying on multivariate extreme value theory have been developed in the past few years for identifying such events/subgroups. This paper exploits this approach much further by means of a novel mixture model that permits to describe the distribution of extremal observations and where the anomaly type $α$ is viewed as a latent variable. One may then take advantage of the model by assigning to any extreme point a posterior probability for each anomaly type $α$, defining implicitly a similarity measure between anomalies. It is explained at length how the latter permits to cluster extreme observations and obtain an informative planar representation of anomalies using standard graph-mining tools. The relevance and usefulness of the clustering and 2-d visual display thus designed is illustrated on simulated datasets and on real observations as well, in the aeronautics application domain.
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