# Subexponential-Time Algorithms for Sparse PCA

We study the computational cost of recovering a unit-norm sparse principal component $x \in \mathbb{R}^n$ planted in a random matrix, in either the Wigner or Wishart spiked model (observing either $W + \lambda xx^\top$ with $W$ drawn from the Gaussian orthogonal ensemble, or $N$ independent samples from $\mathcal{N}(0, I_n + \beta xx^\top)$, respectively). Prior work has shown that when the signal-to-noise ratio ($\lambda$ or $\beta\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = \rho$, it is possible to recover $x$ in polynomial time if $\rho \lesssim 1/\sqrt{n}$. While it is possible to recover $x$ in exponential time under the weaker condition $\rho \ll 1$, it is believed that polynomial-time recovery is impossible unless $\rho \lesssim 1/\sqrt{n}$. We investigate the precise amount of time required for recovery in the "possible but hard" regime $1/\sqrt{n} \ll \rho \ll 1$ by exploring the power of subexponential-time algorithms, i.e., algorithms running in time $\exp(n^\delta)$ for some constant $\delta \in (0,1)$. For any $1/\sqrt{n} \ll \rho \ll 1$, we give a recovery algorithm with runtime roughly $\exp(\rho^2 n)$, 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 $\exp(\rho n)$-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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