Min-Max Grassmannian Optimization for Online Subspace Tracking

TL;DR

This paper introduces GeRoST, a robust, efficient online subspace tracking algorithm based on a min-max optimization over Grassmannian manifolds, with proven robustness guarantees and practical validation.

Contribution

It proposes a novel min-max optimization framework for robust subspace tracking on Grassmannians, providing a closed-form solution for worst-case subspaces and an efficient algorithm.

Findings

Validated on linear time-varying system tracking

Effective in online foreground-background separation in videos

Offers robustness guarantees with computational efficiency

Abstract

This paper discusses robustness guarantees for online tracking of time-varying subspaces from noisy data. Building on recent work in optimization over a Grassmannian manifold, we introduce a new approach for robust subspace tracking by modeling data uncertainty in a Grassmannian ball. The robust subspace tracking problem is cast into a min-max optimization framework, for which we derive a closed-form solution for the worst-case subspace, enabling a geometric robustness adjustment that is both analytically tractable and computationally efficient, unlike iterative convex relaxations. The resulting algorithm, GeRoST (Geometrically Robust Subspace Tracking), is validated on two case studies: tracking a linear time-varying system and online foreground-background separation in video.