标题： 基于协方差逆的梯度优化线性判别分析 显示英文标题

标题： Linear Discriminant Analysis with Gradient Optimization on Covariance Inverse

摘要： 线性判别分析（LDA）是统计模式识别和分类的基本方法，在高斯假设下可达到贝叶斯最优性。然而，众所周知，由于协方差估计的不稳定性，经典LDA在高维情况下可能会表现不佳。本文提出了一种新的方法——梯度优化LDA（LDA-GO），该方法通过梯度下降直接优化逆协方差矩阵。算法通过Cholesky分解参数化逆协方差矩阵，引入低秩扩展以降低计算复杂度，并采用多初始值策略，包括单位矩阵初始化和从经典LDA估计值热启动。通过广泛的多元模拟和真实数据实验验证了LDA-GO的有效性。 显示英文摘要

摘要： Linear discriminant analysis (LDA) is a fundamental method in statistical pattern recognition and classification, achieving Bayes optimality under Gaussian assumptions. However, it is well-known that classical LDA may struggle in high-dimensional settings due to instability in covariance estimation. In this work, we propose LDA with gradient optimization (LDA-GO), a new approach that directly optimizes the inverse covariance matrix via gradient descent. The algorithm parametrizes the inverse covariance matrix through Cholesky factorization, incorporates a low-rank extension to reduce computational complexity, and considers a multiple-initialization strategy, including identity initialization and warm-starting from the classical LDA estimates. The effectiveness of LDA-GO is demonstrated through extensive multivariate simulations and real-data experiments.