Reduced-rank linear discriminant analysis (RRLDA) is a foundational method of dimension reduction for classification that has been useful in a wide range of applications. The goal is to identify an optimal subspace to project the observations onto that simultaneously maximizes between-group variation while minimizing within-group differences. The solution is straight forward when the number of observations is greater than the number of features but computational difficulties arise in both the high-dimensional setting, where there are more features than there are observations, and when the data are very large. Many works have proposed solutions for the high-dimensional setting and frequently involve additional assumptions or tuning parameters. We propose a fast and simple iterative algorithm for both classical and high-dimensional RRLDA on large data that is free from these additional requirements and that comes with guarantees. We also explain how RRLDA-RK provides implicit regularization towards the least norm solution without explicitly incorporating penalties. We demonstrate our algorithm on real data and highlight some results.

翻译：降秩线性判别分析（RRLDA）是一种用于分类的降维基础方法，已在广泛应用中证明其有效性。其目标是识别一个最优子空间，将观测数据投影到该子空间上，以同时最大化组间差异并最小化组内差异。当观测数量大于特征数量时，求解过程较为直接；但在高维设置（特征数量多于观测数量）以及数据规模极大时，会出现计算困难。许多研究已针对高维设置提出了解决方案，这些方案通常涉及额外的假设或调优参数。我们提出了一种快速、简单的迭代算法，适用于大规模数据的经典及高维RRLDA，该算法无需这些额外要求，并具有理论保证。我们还解释了RRLDA-RK如何在不显式引入惩罚项的情况下，隐式地正则化至最小范数解。我们在真实数据上验证了该算法，并重点展示了一些结果。