用最优拟合施佩尔曼簇替代子空间均值，提升聚类精度。
A Granular Grassmannian Clustering Framework via the Schubert Variety of Best Fit
- 以施佩尔曼簇为可训练原型，通过固定方向逼近每个成员
- 在合成、图像、光谱和视频动作数据上实现更高聚类纯度
- 兼容下游分析，适合需要几何结构的子空间聚类任务
在许多分类与聚类任务中，计算数据集或聚类的几何代表（如均值或中位数）很有用。当数据以子空间表示时，这些代表即为格拉斯曼流形或旗流形上的点，其距离由主角几何决定。本文提出一种子空间聚类算法，用可训练的最优拟合施佩尔曼簇（SVBF）代替子空间均值——该子空间在至少一个固定方向上尽可能与每个聚类成员相交。集成至林德-布佐-格雷（LBG）流程后，该SVBF-LBG方案在合成、图像、光谱及视频动作数据上实现了更高的聚类纯度，同时保留了下游分析所需的数学结构。
原文摘要 · Abstract (English)
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.
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