Decoupling Homogeneous spectral offsets in PCA: The Power of Skew-Symmetry

Abstract: Streaming Principal Component Analysis (PCA) without explicit matrix materialization faces a fundamental bottleneck when sequential observations are corrupted by homogeneous spectral offsets and unstructured errors. Traditional stochastic approximation algorithms typically falter in this regime: their fixed-step schedules are heavily penalized by the inflated overall operator norm, whereas adaptive scaling strategies suffer from severe deceleration due to vanishing updates.To circumvent these limitations, we propose a discrete-time optimization framework driven by skew-symmetric generators. By their algebraic nature, these skew-symmetric flows completely filter out uniform diagonal perturbations, guaranteeing exact immunity to the $\sigma_t^2 I$ term at every discrete integration step. Consequently, both the optimization trajectory and the algorithmic stability threshold are governed strictly by the traceless component of the underlying covariance. By leveraging the strict-saddle property of the objective alongside input-to-state stability tools, we establish global convergence and robust sample efficiency against unstructured noise. Furthermore, through a rigorous analysis of the dynamics around degenerate subspaces, we demonstrate rapid saddle-point evasion, culminating in comprehensive finite-time performance guarantees.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Antoine_Patrick_Isabelle_Eric_Ledent1

Submission Number: 8191