Polynomial-in-sparsity quadratic-time sparse PCA

Develop an algorithm for the relaxed sparse PCA certification problem whose running time is d^2\cdot\operatorname{poly}(k), thereby improving the paper’s d^2 k^{O(\log k)}-time guarantee in the highly sparse regime.

Background

The paper studies sparse PCA certification for a positive semidefinite matrix, allowing the algorithm to return either a standard semidefinite-programming witness or a sparse vector whose support is larger than the target sparsity. This bicriteria relaxation enables a quadratic-time algorithm with running time O(d^2+d k^{O(\log k)}) and a subquadratic sample-access variant, but the dependence on the sparsity parameter remains quasi-polynomial.

The unresolved issue is whether the ambient-dimension dependence can remain quadratic while the sparsity dependence is reduced to polynomial. Such an improvement would strengthen the paper’s fast certification results and could also yield faster robust sparse estimation procedures through the paper’s reduction from robust mean estimation to sparse PCA certification.