Reinforced Angle-Based Multicategory Support Vector Machines

The Support Vector Machine (SVM) is a very popular classification tool with many successful applications. It was originally designed for binary problems with desirable theoretical properties. Although there exist various Multicategory SVM (MSVM) extensions in the literature, some challenges remain. In particular, most existing MSVMs make use of k classification functions for a k-class problem, and the corresponding optimization problems are typically handled by existing quadratic programming solvers. In this paper, we propose a new group of MSVMs, namely the Reinforced Angle-based MSVMs (RAMSVMs), using an angle-based prediction rule with k − 1 functions directly. We prove that RAMSVMs can enjoy Fisher consistency. Moreover, we show that the RAMSVM can be implemented using the very efficient coordinate descent algorithm on its dual problem. Numerical experiments demonstrate that our method is highly competitive in terms of computational speed, as well as classification prediction performance. Supplemental materials for the article are available online

Robust Model-Free Multiclass Probability Estimation

Classical statistical approaches for multiclass probability estimation are typically based on regression techniques such as multiple logistic regression, or density estimation approaches such as linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). These methods often make certain assumptions on the form of probability functions or on the underlying distributions of subclasses. In this article, we develop a model-free procedure to estimate multiclass probabilities based on large-margin classifiers. In particular, the new estimation scheme is employed by solving a series of weighted large-margin classifiers and then systematically extracting the probability information from these multiple classification rules. A main advantage of the proposed probability estimation technique is that it does not impose any strong parametric assumption on the underlying distribution and can be applied for a wide range of large-margin classification methods. A general computational algorithm is developed for class probability estimation. Furthermore, we establish asymptotic consistency of the probability estimates. Both simulated and real data examples are presented to illustrate competitive performance of the new approach and compare it with several other existing methods

Composite multiclass losses

We consider loss functions for multiclass prediction problems. We show when a multiclass loss can be expressed as a “proper composite loss”, which is the composition of a proper loss and a link function. We extend existing results for binary losses to multiclass losses. We subsume results on “classification calibration” by relating it to properness. We determine the stationarity condition, Bregman representation, order-sensitivity, and quasi-convexity of multiclass proper losses. We then characterise the existence and uniqueness of the composite representation formulti class losses. We show how the composite representation is related to other core properties of a loss: mixability, admissibility and (strong) convexity of multiclass losses which we characterise in terms of the Hessian of the Bayes risk. We show that the simple integral representation for binary proper losses can not be extended to multiclass losses but offer concrete guidance regarding how to design different loss functions. The conclusion drawn from these results is that the proper composite representation is a natural and convenient tool for the design of multiclass loss functions

Unified Binary and Multiclass Margin-Based Classification

The notion of margin loss has been central to the development and analysis of algorithms for binary classification. To date, however, there remains no consensus as to the analogue of the margin loss for multiclass classification. In this work, we show that a broad range of multiclass loss functions, including many popular ones, can be expressed in the relative margin form, a generalization of the margin form of binary losses. The relative margin form is broadly useful for understanding and analyzing multiclass losses as shown by our prior work (Wang and Scott, 2020, 2021). To further demonstrate the utility of this way of expressing multiclass losses, we use it to extend the seminal result of Bartlett et al. (2006) on classification-calibration of binary margin losses to multiclass. We then analyze the class of Fenchel-Young losses, and expand the set of these losses that are known to be classification-calibrated

Multisensor systems and flood risk management. Application to the Danube Delta using radar and hyperspectral imagery

International audienceAt the beginning of the 21st century, flood risk still represents a major world threat ( 60% of natural disasters are initiated by storms ) and the climate warming might even accentuate this phenomenon in the future. In Europe, despite all the policies in place and the measures taken during the past decades, large floods have occurred almost every year. The news regularly confirms this reality and the serious threat posed by flood risks in Europe. This paper presents an application to the Danube Delta exploiting radar imagery ENVISAT/ASAR and hyperspectral imagery CHRIS/PROBA for mapping flooded and floodable areas during the events of spring 2006. The uses of multisensor systems, such as radar and hyperspectral imagers, contribute to a better comprehension of floods in this wetland, their impacts, and risk management and sustainable development in the delta. In the section Risk management, this paper approaches the methodological aspects related to the characterization of the flood hazard whereas in the section Forecasting we will focus on the knowledge and modeling of the Land cover. The method of kernels, particularly adapted to the highlighting of the special-temporal variations - Support Vector Machine - and the methods based on the principle of the vague logic ( object-oriented classifications ) will be implemented so as to obtain the information plan of the spatial data.En ce début de 21e siècle, le risque d'inondation constitue encore le risque majeur au monde ( avec les tempêtes, les inondations représentent 60% des catastrophes naturelles ) et le réchauffement climatique pourrait encore renforcer ce phénomène à l'avenir. En Europe, malgré toutes les politiques et les mesures prises, au cours des dernières décennies, de grandes inondations ont lieu quasiment chaque année. Les actualités confirment régulièrement la réalité et la prégnance du risque d'inondation en Europe. Cet article présente une application concernant le risque d'inondation durant les événements du printemps 2006 dans le delta du Danube en exploitant des images radar ENVISAT/ASAR et l'imagerie hyperspectrale CHRIS/PROBA en matière d'analyse et de cartographie des zones inondées et de la classe de l'inondable. L'utilisation couplée des techniques spatiales ( radar et hyperspectrale ) pourrait contribuer à une meilleure compréhension des phénomènes liés aux inondations dans le Delta du Danube, ainsi qu'à la gestion de ce risque dans le delta et à son développement durable. Dans la partie Gestion du risque, ce travail aborde des aspects méthodologiques liés à la caractérisation de l'aléa de l'inondation tandis que dans la partie Prévision, la connaissance et la modélisation de l'Occupation du sol seront abordés. Des méthodes des noyaux ( kernels ), adaptées en particulier à la mise en évidence des variations spatio-temporelles - Suport Vector Machine - ainsi que des méthodes basées sur le principe de la logique floue ( classifications orientées objet ) sont mis en place afin d'obtenir le plan d'information des données spatiales

Composite Multiclass Losses

We consider loss functions for multiclass prediction problems. We show when a multiclass loss can be expressed as a "proper composite loss", which is the composition of a proper loss and a link function. We extend existing results for binary losses to mu