Linear and Quadratic Discriminant Analysis (LDA and QDA) are well-known
classical methods but can heavily suffer from non-Gaussian distributions and/or
contaminated datasets, mainly because of the underlying Gaussian assumption
that is not robust. This paper studies the robustness to scale changes in the
data of a new discriminant analysis technique where each data point is drawn by
its own arbitrary Elliptically Symmetrical (ES) distribution and its own
arbitrary scale parameter. Such a model allows for possibly very heterogeneous,
independent but non-identically distributed samples. The new decision rule
derived is simple, fast, and robust to scale changes in the data compared to
other state-of-the-art method

研究一种新的辨别分析技术，可以应对非高斯分布和数据污染，采用椭圆对称分布和任意比例参数建模，从而实现抗尺度变化的简单、快速、鲁棒的决策规则

FEMDA：一种强大而灵活的分类方法

FEMDA: Une méthode de classification robuste et flexible

In this paper, we study high-dimensional sparse Quadratic Discriminant
Analysis (QDA) and aim to establish the optimal convergence rates for the
classification error. Minimax lower bounds are established to demonstrate the
necessity of structural assumptions such as sparsity conditions on the
discriminating direction and differential graph for the possible construction
of consistent high-dimensional QDA rules. We then propose a classification
algorithm called SDAR using constrained convex optimization under the sparsity
assumptions. Both minimax upper and lower bounds are obtained and this
classification rule is shown to be simultaneously rate optimal over a
collection of parameter spaces, up to a logarithmic factor. Simulation studies
demonstrate that SDAR performs well numerically. The algorithm is also
illustrated through an analysis of prostate cancer data and colon tissue data.
The methodology and theory developed for high-dimensional QDA for two groups in
the Gaussian setting are also extended to multi-group classification and to
classification under the Gaussian copula model.

本文研究高维稀疏二次判别分析 (QDA)，旨在建立最优的收敛速率以实现分类误差。我们提出了一种分类算法 SDAR，使用基于稀疏假设下的约束凸优化。该算法的最小化上下界都被得到。模拟研究证明，SDAR 在数值上表现良好。我们还将此方法论扩展到多组分类和高斯椎体模型分类。

高维稀疏二次判别分析的凸优化方法

A Convex Optimization Approach to High-Dimensional Sparse Quadratic  Discriminant Analysis

This tutorial explains Linear Discriminant Analysis (LDA) and Quadratic
Discriminant Analysis (QDA) as two fundamental classification methods in
statistical and probabilistic learning. We start with the optimization of
decision boundary on which the posteriors are equal. Then, LDA and QDA are
derived for binary and multiple classes. The estimation of parameters in LDA
and QDA are also covered. Then, we explain how LDA and QDA are related to
metric learning, kernel principal component analysis, Mahalanobis distance,
logistic regression, Bayes optimal classifier, Gaussian naive Bayes, and
likelihood ratio test. We also prove that LDA and Fisher discriminant analysis
are equivalent. We finally clarify some of the theoretical concepts with
simulations we provide.

该论文介绍了线性判别分析和二次判别分析作为统计和概率学习中的两种基本分类方法，涵盖了决策边界优化、二元和多元类别的 LDA 和 QDA、LDA 和 QDA 与度量学习、核 PCA、马氏距离，Logistic 回归、朴素贝叶斯、最大似然比检验的联系，证明了 LDA 和 Fisher 判别分析是等价的，并通过模拟进行了一些理论的澄清。