# 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 (n^δ) for some constant δ∈ (0,1). For any 1/√(n)≪ρ≪ 1, we give a recovery algorithm with runtime roughly (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 (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.

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