Subexponential-Time Algorithms for Sparse PCA

We study the computational cost of recovering a unit-norm sparse principal component x ∈ℝ^n planted in a random matrix, in either the Wigner or Wishart spiked model (observing either W + λ xx^⊤ with W drawn from the Gaussian orthogonal ensemble, or N independent samples from 𝒩(0, I_n + β xx^⊤ ) , respectively). Prior work has shown that when the signal-to-noise ratio ( λ or β√(N/n) , respectively) is a small constant and the fraction of nonzero entries in the planted vector is ‖ x‖ _0 / n = ρ , it is possible to recover x in polynomial time if ρ≲ 1/√(n) . While it is possible to recover x in exponential time under the weaker condition ρ≪ 1 , it is believed that polynomial-time recovery is impossible unless ρ≲ 1/√(n) . We investigate the precise amount of time required for recovery in the “possible but hard” regime 1/√(n)≪ρ≪ 1 by exploring the power of subexponential-time algorithms, i.e., algorithms running in time exp (n^δ ) for some constant δ∈ (0,1) . For any 1/√(n)≪ρ≪ 1 , we give a recovery algorithm with runtime roughly exp (ρ ^2 n) , demonstrating a smooth tradeoff between sparsity and runtime. Our family of algorithms interpolates smoothly between two existing algorithms: the polynomial-time diagonal thresholding algorithm and the exp (ρ n) -time exhaustive search algorithm. Furthermore, by analyzing the low-degree likelihood ratio, we give rigorous evidence suggesting that the tradeoff achieved by our algorithms is optimal.