论文标题
通过新颖的基于原型的可解释解决方案重新访问经典多类线性判别分析
Revisiting Classical Multiclass Linear Discriminant Analysis with a Novel Prototype-based Interpretable Solution
论文作者
论文摘要
线性判别分析（LDA）是特征提取和降低维度降低的基本方法。尽管有许多变体，但古典LDA具有其自身的重要性，因为它是人类有关统计模式识别的基石。对于包含C簇的数据集，这是大多数C-1功能的LDA提取物的经典解决方案。在这里，我们为称为LDA ++的经典LDA介绍了一种新颖的解决方案，该解决方案产生C特征，每种都可以解释为测量与一个群集的相似性。这个新颖的解决方案桥梁降低了维度的降低和多类分类。具体而言，我们证明，对于均质的高斯数据，在某些温和条件下，线性多类分类器的最佳权重也为LDA提供了最佳解决方案。此外，我们表明LDA ++揭示了有关LDA的一些重要新事实，这些事实显着改变了我们在引入75年后对经典多类LDA的理解。我们为病例提供了LDA ++的完整数值解决方案1）当可以明确构造散点矩阵时，2）当构造散点矩阵时，不可行，3）内核扩展。
Linear discriminant analysis (LDA) is a fundamental method for feature extraction and dimensionality reduction. Despite having many variants, classical LDA has its own importance, as it is a keystone in human knowledge about statistical pattern recognition. For a dataset containing C clusters, the classical solution to LDA extracts at most C-1 features. Here, we introduce a novel solution to classical LDA, called LDA++, that yields C features, each interpretable as measuring similarity to one cluster. This novel solution bridges dimensionality reduction and multiclass classification. Specifically, we prove that, for homoscedastic Gaussian data and under some mild conditions, the optimal weights of a linear multiclass classifier also make an optimal solution to LDA. In addition, we show that LDA++ reveals some important new facts about LDA that remarkably changes our understanding of classical multiclass LDA after 75 years of its introduction. We provide a complete numerical solution for LDA++ for the cases 1) when the scatter matrices can be constructed explicitly, 2) when constructing the scatter matrices is infeasible, and 3) the kernel extension.
