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- W4317494823 abstract "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 $$Vert xVert _0 / n = rho $$ , 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." @default.
- W4317494823 created "2023-01-20" @default.
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- W4317494823 date "2023-01-19" @default.
- W4317494823 modified "2023-09-25" @default.
- W4317494823 title "Subexponential-Time Algorithms for Sparse PCA" @default.
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- W4317494823 doi "https://doi.org/10.1007/s10208-023-09603-0" @default.
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