Subexponential-Time Algorithms for Sparse PCA

We study the computational cost of recovering a unit-norm sparse principal component x ∈ ℝn planted in a random matrix, in either the Wigner or Wishart spiked model (observing either W + λ xx^top with W drawn from the Gaussian orthogonal ensemble, or N independent samples from N(0, In + β xx^top), 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 lesssim 1/√n. While it is possible to recover x in exponential time under the weaker condition ρ ll 1, it is believed that polynomial-time recovery is impossible unless ρ lesssim 1/√n. We investigate the precise amount of time required for recovery in the "possible but hard" regime 1/√n ll ρ ll 1 by exploring the power of subexponential-time algorithms, i.e., algorithms running in time exp(nδ) for some constant δ ∈ (0,1). For any 1/√n ll ρ ll 1, we give a recovery algorithm with runtime roughly exp(ρ2 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.