Sharpen the λ1, λ2 dependence in the thresholded Oja error bound

Improve the dependence on the leading and second eigenvalues (λ1 and λ2) in the sin^2 error bound proved for the thresholded-and-renormalized Oja algorithm (Algorithm 1; Theorem 3.1, Convergence of Oja with cardinality support) so that it matches the corresponding minimax lower bounds for sparse principal component estimation established by Vu and Lei (2012) and by Cai, Ma, and Wu (2013).

Background

The paper proves that hard-thresholding and renormalizing the output of Oja’s algorithm achieves a near-optimal sin^2 error of order O(s log d / n) in the high-dimensional streaming setting with an s-sparse leading eigenvector. The bound’s dependence on λ1 and λ2 is described as near-optimal, and the authors explicitly state a desire to improve it.

Classical minimax lower bounds for sparse PCA in the offline setting (Vu and Lei, 2012; Cai, Ma, and Wu, 2013) characterize sharp dependence on problem parameters, including eigenvalues. Aligning the streaming algorithm’s bound with these lower bounds would close a gap in the analysis and yield sharp parameter dependence.