Statistical-Computational Tradeoffs in Mixed Sparse Linear Regression
| Autor: | Arpino, Gabriel, Venkataramanan, Ramji |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | We consider the problem of mixed sparse linear regression with two components, where two real $k$-sparse signals $\beta_1, \beta_2$ are to be recovered from $n$ unlabelled noisy linear measurements. The sparsity is allowed to be sublinear in the dimension, and additive noise is assumed to be independent Gaussian with variance $\sigma^2$. Prior work has shown that the problem suffers from a $\frac{k}{SNR^2}$-to-$\frac{k^2}{SNR^2}$ statistical-to-computational gap, resembling other computationally challenging high-dimensional inference problems such as Sparse PCA and Robust Sparse Mean Estimation; here $SNR$ is the signal-to-noise ratio. We establish the existence of a more extensive computational barrier for this problem through the method of low-degree polynomials, but show that the problem is computationally hard only in a very narrow symmetric parameter regime. We identify a smooth information-computation tradeoff between the sample complexity $n$ and runtime for any randomized algorithm in this hard regime. Via a simple reduction, this provides novel rigorous evidence for the existence of a computational barrier to solving exact support recovery in sparse phase retrieval with sample complexity $n = \tilde{o}(k^2)$. Our second contribution is to analyze a simple thresholding algorithm which, outside of the narrow regime where the problem is hard, solves the associated mixed regression detection problem in $O(np)$ time with square-root the number of samples and matches the sample complexity required for (non-mixed) sparse linear regression; this allows the recovery problem to be subsequently solved by state-of-the-art techniques from the dense case. As a special case of our results, we show that this simple algorithm is order-optimal among a large family of algorithms in solving exact signed support recovery in sparse linear regression. Comment: To appear in Conference on Learning Theory (COLT) 2023 |
| Databáze: | arXiv |
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