This paper deals with supervised classification and feature selection in high dimensional space. A classical approach is to project data on a low dimensional space and classify by minimizing an appropriate quadratic cost. A strict control on sparsity is moreover obtained by adding an $\ell_1$ constraint, here on the matrix of weights used for projecting the data. Tuning the sparsity bound results in selecting the relevant features for supervised classification. It is well known that using a quadratic cost is not robust to outliers. We cope with this problem by using an $\ell_1$ norm both for the constraint and for the loss function. In this case, the criterion is convex but not gradient Lipschitz anymore. Another second issue is that we optimize simultaneously the projection matrix and the centers used for classification. In this paper, we provide a novel tailored constrained primal-dual method to compute jointly selected features and classifiers. Extending our primal-dual method to other criteria is easy provided that efficient projection (on the dual ball for the loss data term) and prox (for the regularization term) algorithms are available. We illustrate such an extension in the case of a Frobenius norm for the loss term. We provide a convergence proof of our primal-dual method, and demonstrate its effectiveness on three datasets (one synthetic, two from biological data) on which we compare $\ell_1$ and $\ell_2$ costs.
Stochastic Primal Dual Coordinate Method with Non-Uniform Sampling Based on Optimality Violations
Nagoya Institute of Technology（名古屋工业大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所革新智能综合研究中心）；National Institute for Materials Science（国立材料科学研究所）
Random extrapolation for primal-dual coordinate descent
EPFL（洛桑联邦理工学院）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎综合理工学院）
我们引入了一种随机外推的原始-对偶坐标下降法，该方法适应于数据矩阵的稀疏性和目标函数的有利结构。我们的方法只更新原始和对偶变量的一个子集与稀疏的数据，它使用大的步长与密集的数据，保留的好处，为每种情况下设计的具体方法。除了适应稀疏性，我们的方法达到快速收敛的保证在有利的情况下\textit{without any modifications}。特别地，我们证明了度量次正则下的线性收敛性，它适用于强凸-强凹问题和分段线性二次函数。在一般的凸-凹情形下，我们证明了序列的几乎必然收敛和原始-对偶间隙和目标值的最佳次线性收敛速度。数值证据证明了我们的方法在稀疏和密集环境中的最先进的经验性能，匹配和改进了现有的方法。
Online Primal-Dual Algorithms with Predictions for Packing Problems
University Paris-Saclay（巴黎萨克雷大学）；Sorbonne University（巴黎索邦大学）；IBISC（IBISC（需注意：IBISC是巴黎萨克雷大学下属研究机构，此处按原文提取））；CNRS（法国国家科学研究中心）；LIP6（LIP6（巴黎索邦大学下属实验室，此处按原文提取））
Enabling self-supervised learned primal dual with Noise2Inverse
Research Unit of Mathematical Sciences, University of Oulu, Finland（奥卢大学数学科学研究中心，芬兰）；Department of Mathematics and Information Science, Josai University, Japan（立命馆大学数学与信息科学系，日本）；Department of Computer Science, University College London, United Kingdom（伦敦大学学院计算机科学系，英国）
