A Granular Grassmannian Clustering Framework via the Schubert Variety of Best Fit
This work addresses subspace clustering tasks for applications like image and video analysis, but it is incremental as it builds on the Linde-Buzo-Grey pipeline.
The paper tackled the problem of subspace clustering by introducing a trainable prototype called the Schubert Variety of Best Fit (SVBF) to replace subspace means, resulting in improved cluster purity on synthetic, image, spectral, and video action data.
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.