Remove logarithmic overhead and privately estimate the top eigenvalue

Remove the remaining polylogarithmic sample-complexity overhead in the private energy-PCA guarantee of Theorem 5.2 and develop a private procedure to estimate the largest eigenvalue \(\lambda_1\) from samples.

Background

The paper develops a differentially private variant of Oja’s algorithm for gap-free energy PCA under a sub-Gaussian data model. For Gaussian data with publicly known top eigenvalue λ1\lambda_1, the resulting sample complexity matches the rate conjectured in prior work up to logarithmic factors. The authors identify two unresolved issues: eliminating those remaining logarithmic factors and handling the practically important case in which λ1\lambda_1 is not publicly known by estimating it privately from the data.

References

The main outstanding questions left by Theorem~\ref{thm:private_epca} are to remove the remaining polylogarithmic overhead, and to privately estimate $_1$ from samples.

— 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 “Our results,” immediately following Theorem~\ref{thm:private_epca}