Nonparametric Linear Feature Learning in Regression Through Regularisation

Representation learning plays a crucial role in automated feature selection, particularly in the context of high-dimensional data, where non-parametric methods often struggle. In this study, we focus on supervised learning scenarios where the pertinent information resides within a lower-dimensional linear subspace of the data, namely the multi-index model. If this subspace were known, it would greatly enhance prediction, computation, and interpretation. To address this challenge, we propose a novel method for linear feature learning with non-parametric prediction, which simultaneously estimates the prediction function and the linear subspace. Our approach employs empirical risk minimisation, augmented with a penalty on function derivatives, ensuring versatility. Leveraging the orthogonality and rotation invariance properties of Hermite polynomials, we introduce our estimator, named RegFeaL. By utilising alternative minimisation, we iteratively rotate the data to improve alignment with leading directions and accurately estimate the relevant dimension in practical settings. We establish that our method yields a consistent estimator of the prediction function with explicit rates. Additionally, we provide empirical results demonstrating the performance of RegFeaL in various experiments.Comment: 42 pages, 5 figure

Nonparametric Linear Feature Learning in Regression Through Regularisation

42 pages, 5 figuresRepresentation learning plays a crucial role in automated feature selection, particularly in the context of high-dimensional data, where non-parametric methods often struggle. In this study, we focus on supervised learning scenarios where the pertinent information resides within a lower-dimensional linear subspace of the data, namely the multi-index model. If this subspace were known, it would greatly enhance prediction, computation, and interpretation. To address this challenge, we propose a novel method for linear feature learning with non-parametric prediction, which simultaneously estimates the prediction function and the linear subspace. Our approach employs empirical risk minimisation, augmented with a penalty on function derivatives, ensuring versatility. Leveraging the orthogonality and rotation invariance properties of Hermite polynomials, we introduce our estimator, named RegFeaL. By utilising alternative minimisation, we iteratively rotate the data to improve alignment with leading directions and accurately estimate the relevant dimension in practical settings. We establish that our method yields a consistent estimator of the prediction function with explicit rates. Additionally, we provide empirical results demonstrating the performance of RegFeaL in various experiments

High-dimensional changepoint estimation with heterogeneous missingness

We propose a new method for changepoint estimation in partially-observed, high-dimensional time series that undergo a simultaneous change in mean in a sparse subset of coordinates. Our first methodological contribution is to introduce a 'MissCUSUM' transformation (a generalisation of the popular Cumulative Sum statistics), that captures the interaction between the signal strength and the level of missingness in each coordinate. In order to borrow strength across the coordinates, we propose to project these MissCUSUM statistics along a direction found as the solution to a penalised optimisation problem tailored to the specific sparsity structure. The changepoint can then be estimated as the location of the peak of the absolute value of the projected univariate series. In a model that allows different missingness probabilities in different component series, we identify that the key interaction between the missingness and the signal is a weighted sum of squares of the signal change in each coordinate, with weights given by the observation probabilities. More specifically, we prove that the angle between the estimated and oracle projection directions, as well as the changepoint location error, are controlled with high probability by the sum of two terms, both involving this weighted sum of squares, and representing the error incurred due to noise and the error due to missingness respectively. A lower bound confirms that our changepoint estimator, which we call 'MissInspect', is optimal up to a logarithmic factor. The striking effectiveness of the MissInspect methodology is further demonstrated both on simulated data, and on an oceanographic data set covering the Neogene period

Lorsque des baux à cheptel ne tournent pas comme il était convenu…

Ce document isolé est une sentence provisoire d’un procès se déroulant devant la juridiction seigneuriale de Bécon en 1689-1690. Le procès est mené par le seul Marin Guérard, « séneschal de la baronnie de Bescon », sans mention d’autres officiers de justice, sinon le sergent Chalain. Il n’y a aucun indice de dialogue entre le juge et un « procureur fiscal ». La cause a été introduite les 14 et 16 décembre 1689 et la sentence éditée infra est du 2 juin 1690. Elle renvoie encore à une audience ..

Les justices locales

Prolongeant le colloque d'Angers Les Justices de Village. Administration et justice locales de la fin du Moyen Âge à La Révolution… (PUR, 2003) cette publication varie les lieux, les échelles et la chronologie. Dans l'ancienne France, il était admis qu'une police assez « bonne » éviterait forcément de recourir à « rigueur de justice ». L'étude de la justice de proximité est donc toujours associée à la « police » ou administration. Mais prenant le contrepied de Charles Loyseau qui dans son Discours de l'abus… (1603) avait construit sa critique des « très mauvaises Justices de Village » par un jeu de miroir avec les « amples Justices des Villes », le présent volume les rapproche et atteint toutes les espèces de juges de l'Ancien Régime (seigneuriaux, municipaux et royaux) ainsi que certains magistrats post-révolutionnaires. Les regards d'historiens des Facultés de Droit et des Lettres ont été croisés pour étudier des sociétés rurales diverses, ainsi que des villes de toutes dimensions, depuis des chefs-lieux de l'Anjou et de Haute-Auvergne jusqu'à des villes moyennes et capitales. Transcendant aussi la césure révolutionnaire, l'ouvrage approfondit l'hypothèse d'une succession entre la justice seigneuriale (abolie en 1789) et la justice de paix (instituée en 1790). À ce degré de proximité « Vautil mieux s'arranger que plaider ? » Sont donc examinés les renvois des juges pour « accommodements » par les curés sous l'Ancien Régime, diverses formes d'« arbitrages », la procédure de « conciliation » par les juges de paix et la question rarement étudiée de la justice et police des maires. Qu'ont fait ces magistrats de leurs nouveaux pouvoirs de justice et police ?À Jacques-Guy Petit qui a créé l'HIRES et à Jacques Maillard, avec toute mon amitié