For high-dimensional classification, interpolation of training data manifests as the data piling phenomenon, in which linear projections of data vectors from each class collapse to a single value. Recent research has revealed an additional phenomenon known as the ‘second data piling’ for independent test data in binary classification, providing a theoretical understanding of asymptotically perfect classification. This paper extends these findings to multi-category classification and provides a comprehensive characterization of the double data piling phenomenon. We define the maximal data piling subspace, which maximizes the sum of pairwise distances between piles of training data in multi-category classification. Furthermore, we show that a second data piling subspace that induces data piling for independent data exists and can be consistently estimated by projecting the negatively-ridged discriminant subspace onto an estimated ‘signal’ subspace. By leveraging this second data piling phenomenon, we propose a bias-correction strategy for class assignments, which asymptotically achieves perfect classification. The present research sheds light on benign overfitting and enhances the understanding of perfect multi-category classification of high-dimensional discrimination with a help of high-dimensional asymptotics.

对于高维分类，训练数据的插值表现为数据堆积现象，其中每个类别的数据向量的线性投影崩溃为单个值。最近的研究揭示了二元分类中独立测试数据的另一种现象，称为“第二数据堆积”，为渐近完美分类提供了理论理解。本文将这些发现扩展到多类别分类，并提供了双重数据堆积现象的全面表征。我们定义最大数据堆积子空间，它最大化多类别分类中训练数据堆之间的成对距离之和。此外，我们表明存在导致独立数据的数据堆积的第二数据堆积子空间，并且可以通过将负脊判别子空间投影到估计的“信号”子空间来一致地估计。通过利用第二个数据堆积现象，我们提出了一种用于类别分配的偏差校正策略，渐近地实现了完美的分类。目前的研究揭示了良性过度拟合，并在高维渐近的帮助下增强了对高维判别的完美多类别分类的理解。