Robust classification via MOM minimization

We present an extension of Vapnik's classical empirical risk minimizer (ERM) where the empirical risk is replaced by a median-of-means (MOM) estimator, the new estimators are called MOM minimizers. While ERM is sensitive to corruption of the dataset for many classical loss functions used in classification, we show that MOM minimizers behave well in theory, in the sense that it achieves Vapnik's (slow) rates of convergence under weak assumptions: data are only required to have a finite second moment and some outliers may also have corrupted the dataset. We propose an algorithm inspired by MOM minimizers. These algorithms can be analyzed using arguments quite similar to those used for Stochastic Block Gradient descent. As a proof of concept, we show how to modify a proof of consistency for a descent algorithm to prove consistency of its MOM version. As MOM algorithms perform a smart subsampling, our procedure can also help to reduce substantially time computations and memory ressources when applied to non linear algorithms. These empirical performances are illustrated on both simulated and real datasets

One-Class Classification: Taxonomy of Study and Review of Techniques

One-class classification (OCC) algorithms aim to build classification models when the negative class is either absent, poorly sampled or not well defined. This unique situation constrains the learning of efficient classifiers by defining class boundary just with the knowledge of positive class. The OCC problem has been considered and applied under many research themes, such as outlier/novelty detection and concept learning. In this paper we present a unified view of the general problem of OCC by presenting a taxonomy of study for OCC problems, which is based on the availability of training data, algorithms used and the application domains applied. We further delve into each of the categories of the proposed taxonomy and present a comprehensive literature review of the OCC algorithms, techniques and methodologies with a focus on their significance, limitations and applications. We conclude our paper by discussing some open research problems in the field of OCC and present our vision for future research.Comment: 24 pages + 11 pages of references, 8 figure

Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario

A variety of methods is available to quantify uncertainties arising with\-in the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond

Uncovering Causality from Multivariate Hawkes Integrated Cumulants

We design a new nonparametric method that allows one to estimate the matrix of integrated kernels of a multivariate Hawkes process. This matrix not only encodes the mutual influences of each nodes of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix without any parametric modeling and estimation of the kernels themselves. A consequence is that it can give an estimation of causality relationships between nodes (or users), based on their activity timestamps (on a social network for instance), without knowing or estimating the shape of the activities lifetime. For that purpose, we introduce a moment matching method that fits the third-order integrated cumulants of the process. We show on numerical experiments that our approach is indeed very robust to the shape of the kernels, and gives appealing results on the MemeTracker database