🤖 AI Summary
This work addresses the challenge of efficiently maintaining low-rank approximations of large-scale dynamic matrices—such as financial return panels—where conventional singular value decomposition (SVD) is computationally prohibitive and existing incremental methods struggle to balance accuracy, subspace stability, and efficiency. The authors propose a unified incremental SVD framework that supports row/column appending and rank-1 updates, introducing projection-based explicit update rules that reformulate Brand’s heuristic rank truncation into a quantifiable operation. For the first time, refresh scheduling is treated as a core design dimension, integrating principal angle tracking, error-ratio monitoring, and adaptive rank selection to systematically evaluate periodic and threshold-driven refresh strategies. Experiments on synthetic data streams and multi-asset ETF factor models demonstrate that, with appropriate rank and refresh frequency settings, the method achieves over 95% of full SVD accuracy at minimal computational overhead, making it suitable for high-frequency, batch-infeasible scenarios.
📝 Abstract
Return panels, covariances, and large feature matrices evolve one observation or one entry at a time, yet downstream models require an up-to-date low-rank factorization $A_t \approx U_t Σ_t V_t^\top$ on every tick -- a regime where full SVD is prohibitive and existing alternatives sacrifice either singular vectors, singular values, or long-horizon stability. We present a practical, metric-driven study of Brand-style incremental SVD, built around a unified engine that handles row appends, column appends, rank-1 entry updates, and metrics tracking within a single framework, with two core contributions. For rank-1 entry updates, we derive an explicit projection-based rule $U'Σ'(V')^\top = P_U(\widehat{A} + δ\,e_ie_j^\top)P_V$ that keeps rank fixed while discarding only the out-of-subspace remainder in a quantifiable way, turning Brand's rank-suppression heuristic into an operational scheme. We then treat refresh scheduling as a first-class design axis, systematically comparing periodic, error-threshold, angle-threshold, and adaptive-rank policies on the accuracy-latency frontier. A unified framework tracks error ratios, principal angles, explained variance, and per-update runtime on long synthetic streams and a multi-asset ETF factor model for covariance and portfolio-risk estimation. With a sensible rank and refresh cadence, incremental SVD matches full-SVD accuracy within a few percent at a fraction of the cost, scaling to high-frequency regimes where batch SVDs are infeasible.