Extend gap-free general-update streaming PCA to k-PCA

Extend the gap-free streaming principal component analysis guarantees established for one component under general matrix-valued updates to the \(k\)-PCA problem for every \(k>1\), without imposing rank-one update restrictions or eigengap assumptions.

Background

The paper proves near-optimal gap-free guarantees for one-component streaming PCA using Oja’s algorithm with general matrix-valued stochastic updates. The related-work discussion notes that existing results for kk-PCA occupy separate regimes: one result is gap-free but restricted to rank-one updates, while another permits general updates but requires an eigengap. The authors therefore leave unresolved the analogous result for k>1k>1 that simultaneously allows general updates and avoids eigengap assumptions.

References

We leave open the analogous question for $k$-PCA for $k > 1$, where a similar situation holds in the current literature: gave a gap-free result for $k$-PCA under rank-one updates, and removed the rank restriction, but used an eigengap.

— Gap-Free Streaming PCA Beyond Rank-One Updates: Near-Optimal Rates and Applications to Differential Privacy  (2609.26508 - Gu et al., 22 Sep 2026) in Section 1, subsection “Related work,” paragraph “Streaming and gap-free PCA”