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Nearly Sharp Sufficient Conditions on Exact Sparsity Pattern Recovery

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1 Author(s)
Rahnama Rad, K. ; Dept. of Stat., Columbia Univ., New York, NY, USA

Consider the n-dimensional vector y=Xβ+ε where β ∈ BBRp has only k nonzero entries and ε ∈ BBRn is a Gaussian noise. This can be viewed as a linear system with sparsity constraints corrupted by noise, where the objective is to estimate the sparsity pattern of β given the observation vector y and the measurement matrix X. First, we derive a nonasymptotic upper bound on the probability that a specific wrong sparsity pattern is identified by the maximum-likelihood estimator. We find that this probability depends (inversely) exponentially on the difference of ||Xβ||2 and the l2 -norm of Xβ projected onto the range of columns of X indexed by the wrong sparsity pattern. Second, when X is randomly drawn from a Gaussian ensemble, we calculate a nonasymptotic upper bound on the probability of the maximum-likelihood decoder not declaring (partially) the true sparsity pattern. Consequently, we obtain sufficient conditions on the sample size n that guarantee almost surely the recovery of the true sparsity pattern. We find that the required growth rate of sample size n matches the growth rate of previously established necessary conditions.

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Information Theory, IEEE Transactions on  (Volume:57 ,  Issue: 7 )