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Min-Max Grassmannian Optimization for Online Subspace Tracking

Shreyas Bharadwaj, Bamdev Mishra, Cyrus Mostajeran, Alberto Padoan, Jeremy Coulson, Ravi Banavar

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.

Min-Max Grassmannian Optimization for Online Subspace Tracking

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.

Paper Structure

This paper contains 13 sections, 9 theorems, 38 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Notation
  4. Grassmannian Geometry
  5. PROBLEM STATEMENT
  6. ROBUST SUBSPACE TRACKING
  7. Algorithm
  8. Robustness Guarantees
  9. Convergence Guarantees
  10. NUMERICAL EXPERIMENTS
  11. Case 1: Online System Identification
  12. Case 2: Online Foreground-Background Separation
  13. CONCLUSION

Key Result

Lemma D.1

Consider the matrix $B_t$ as defined in eq:bmatrix. Then the following statements hold. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 3: Relative prediction error of system models estimated by GeRoST and GREAT. The bold lines represent the average over 50 trajectories.
  • Figure 4: Tracking error of the estimated background subspace.
  • Figure 5: ROC curve for foreground-background separation. The curve plots the True Positive Rate (correctly identified foreground pixels) against the False Positive Rate (background pixels incorrectly identified as foreground) across a range of detection thresholds.
  • Figure 6: Evolution of $\rho_t$ and $\lambda^*$ over time. Capping the instantaneous noise-to-signal ratio limits the ball radius $\rho_t$ during occlusion $(t > 50)$, required to separate the sparse foreground from the background subspace.
  • Figure 7: Qualitative comparison of foreground-background separation at frame 250. While the causal nature of online tracking introduces ghosting artifacts in all methods, GeRoST resists severely absorbing the moving outlier into the background subspace, unlike GRASTA and GREAT.

Theorems & Definitions (14)

  • Remark C.1: On the choice of $d$
  • Lemma D.1: Spectral Gap
  • Theorem D.2: Inner Maximization
  • Lemma D.3: Riemannian Gradient
  • Remark D.4: Computational complexity
  • Theorem D.5: Worst-case approximate error bound
  • Corollary D.6
  • Remark D.7: Minimum Uncertainty Radius
  • Lemma D.8
  • Lemma D.9: Single Step Descent
  • ...and 4 more