A Granular Grassmannian Clustering Framework via the Schubert Variety of Best Fit
Karim Salta, Michael Kirby, Chris Peterson
TL;DR
This work tackles subspace clustering on Grassmann manifolds by replacing classical subspace means with a trainable Schubert-based prototype, defined via incidence constraints. The SVBF–LBG algorithm nests a standard LBG loop with an inner SVBF optimization that learns $\mathbf{K}^* = \arg \min_{[\mathbf K]\in \operatorname{Gr}(k,n)} \sum_{i=1}^p \sin^2 \theta_1(\mathbf X_i,\mathbf K)$, leveraging the incidence set $\Omega_{c,k,l}(W)$ to guide subspace fitting. Across synthetic, MNIST, hyperspectral (Indian Pines), and video (UCF11) datasets, SVBF–LBG achieves higher median cluster purity than flag mean or flag median, particularly in incidence-rich scenarios, while preserving manifold structure for downstream analysis. Limitations include a fixed prototype dimension $k$, a non-adaptive stopping rule, and a need for GPU acceleration; future work proposes adaptive directionality in SVBF constraints and convergence analysis, with code available at the authors’ GitHub repository.
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.