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 $n^{-1/3}$-approximation; 2) NP-hardness of approximation to within $(1-varepsilon)$, for some small constant $varepsilon > 0$; 3) SSE-hardness of approximation to within any constant factor; and 4) an $expexpleft(Omegaleft(sqrt{log log n}right)right)$ (quasi-quasi-polynomial) gap for the standard semidefinite program.
We prove that for every $epsilon>0$ and predicate $P:{0,1}^krightarrow {0,1}$ that supports a pairwise independent distribution, there exists an instance $mathcal{I}$ of the $mathsf{Max}P$ constraint satisfaction problem on $n$ variables such that no assignment can satisfy more than a $tfrac{|P^{-1}(1)|}{2^k}+epsilon$ fraction of $mathcal{I}$s constraints but the degree $Omega(n)$ Sum of Squares semidefinite programming hierarchy cannot certify that $mathcal{I}$ is unsatisfiable. Similar results were previously only known for weaker hierarchies.
We prove super-polynomial lower bounds on the size of linear programming relaxations for approximati
