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- W2965893499 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 $betasqrt{N/n}$, 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." @default.
- W2965893499 created "2019-08-13" @default.
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- W2965893499 date "2019-07-26" @default.
- W2965893499 modified "2023-10-03" @default.
- W2965893499 title "Subexponential-Time Algorithms for Sparse PCA" @default.
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- W2965893499 doi "https://doi.org/10.48550/arxiv.1907.11635" @default.
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