Convergence and accuracy complexity of SuperPCA

Establish the convergence rate of the SuperPCA algorithm and determine its complexity for achieving a prescribed final accuracy, including a theoretical justification of the observed O(1/√N) decay and characterization of the dependence of its constant factor on spectral gaps.

Background

SuperPCA starts from a fixed candidate subspace and uses N additional subsampled data vectors to solve a sketched, regularized least-squares problem, producing refined estimates of the desired principal components. Numerical experiments indicate that, for a fixed candidate subspace, the sine of the angle between the estimated and population principal components decreases approximately as O(1/√N).

The paper explicitly states that the convergence behavior has not been proved theoretically. It also observes that the constant in the empirical rate probably depends on a spectral gap, leaving both the convergence analysis and the measurement/ computational complexity required for a target accuracy unresolved.

References

Another question is the study of the convergence of the SuperPCA algorithm, or the complexity of the method for a desired final accuracy. For a fixed candidate subspace we observe a $\mathcal{O}(1/\sqrt{N})$ decrease of the sine of the angle between the desired population principal components and the estimated principal components (where $N$ is the number of subsampled data vectors), with a constant factor that probably depends on some spectral gap, but we leave the theoretical proof for future research.

— SuperPCA: subspace analysis and an efficient algorithm for high-dimensional PCA  (2609.26406 - Haas et al., 22 Sep 2026) in Section Conclusion