In many classification and clustering tasks, it is useful to compute a geometric representative for a dataset or a cluster, such as a mean or median. When datasets are represented by subspaces, these representatives become points on the Grassmann or flag manifold, with distances induced by their geometry, often via principal angles. We introduce a subspace clustering algorithm that replaces subspace means with a trainable prototype defined as a Schubert Variety of Best Fit (SVBF) - a subspace that comes as close as possible to intersecting each cluster member in at least one fixed direction. Integrated in the Linde-Buzo-Grey (LBG) pipeline, this SVBF-LBG scheme yields improved cluster purity on synthetic, image, spectral, and video action data, while retaining the mathematical structure required for downstream analysis.

翻译：在许多分类与聚类任务中，为数据集或簇计算几何代表元（如均值或中位数）具有重要价值。当数据集由子空间表示时，这些代表元成为格拉斯曼流形或旗流形上的点，其距离由流形几何（通常通过主角）所诱导。本文提出一种子空间聚类算法，该算法将子空间均值替换为可训练的原型，该原型被定义为最优拟合舒伯特簇——一个在至少一个固定方向上尽可能接近与每个簇成员相交的子空间。通过将其集成至Linde-Buzo-Grey（LBG）流程中，所提出的SVBF-LBG方案在合成数据、图像数据、光谱数据及视频动作数据上实现了更高的聚类纯度，同时保持了后续分析所需的数学结构。