A Granular Grassmannian Clustering Framework via the Schubert Variety of Best Fit

TL;DR

This paper introduces a novel subspace clustering method using Schubert varieties of best fit, improving cluster purity by leveraging geometric structures on Grassmann manifolds.

Contribution

It proposes a trainable prototype called SVBF within the LBG framework, enhancing clustering of subspace-represented data with geometric rigor.

Findings

Improved cluster purity on various datasets

Effective integration of Schubert varieties into clustering

Maintains mathematical structure for analysis

Abstract

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.