Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting
<|reference_start|>Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting: The problem of recovering the sparsity pattern of a fixed but unknown vector $\beta^* \in \real^p based on a set of $n$ noisy observations arises in a variety of settings, including subset selection in regression, graphical model selection, signal denoising, compressive sensing, and constructive approximation. Of interest are conditions on the model dimension $p$, the sparsity index $s$ (number of non-zero entries in $\beta^*$), and the number of observations $n$ that are necessary and/or sufficient to ensure asymptotically perfect recovery of the sparsity pattern. This paper focuses on the information-theoretic limits of sparsity recovery: in particular, for a noisy linear observation model based on measurement vectors drawn from the standard Gaussian ensemble, we derive both a set of sufficient conditions for asymptotically perfect recovery using the optimal decoder, as well as a set of necessary conditions that any decoder, regardless of its computational complexity, must satisfy for perfect recovery. This analysis of optimal decoding limits complements our previous work (ARXIV: math.ST/0605740) on sharp thresholds for sparsity recovery using the Lasso ($\ell_1$-constrained quadratic programming) with Gaussian measurement ensembles.<|reference_end|>
@article{wainwright2007information-theoretic,
    title={Information-theoretic limits on sparsity recovery in the
  high-dimensional and noisy setting},
    author={Martin J. Wainwright},
    journal={arXiv preprint arXiv:math/0702301},
    year={2007},
    archivePrefix={arXiv},
    eprint={math/0702301},
    primaryClass={math.ST cs.IT math.IT stat.TH}
}