On the Worst-Case Approximability of Sparse PCA

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1. a simple and efficient algorithm that achieves an n-approximation; 2. NP-hardness of approximation to within (1 − ε), for some small constant ε > 0; 3. SSE-hardness of approximation to within any constant factor; and 4. an exp exp ( Ω (√ log logn )) (“quasi-quasi-polynomial”) gap for the standard semidefinite program.