Abstract
The aim of this paper is to develop strategies to estimate the sparsity degree of a signal from compressive projections, without the burden of recovery. We consider both the noise-free and the noisy settings, and we show how to extend the proposed framework to the case of non-exactly sparse signals. The proposed method employs γ-sparsified random matrices and is based on a maximum likelihood (ML) approach, exploiting the property that the acquired measurements are distributed according to a mixture model whose parameters depend on the signal sparsity. In the presence of noise, given the complexity of ML estimation, the probability model is approximated with a two-component Gaussian mixture (2-GMM), which can be easily learned via expectation-maximization.Besides the design of the method, this paper makes two novel contributions. First, in the absence of noise, sufficient conditions on the number of measurements are provided for almost sure exact estimation in different regimes of behavior, defined by the scaling of the measurements sparsity γ and the signal sparsity. In the presence of noise, our second contribution is to prove that the 2-GMM approximation is accurate in the large system limit for a proper choice of γ parameter. Simulations validate our predictions and show that the proposed algorithms outperform the state-of-the-art methods for sparsity estimation. Finally, the estimation strategy is applied to non-exactly sparse signals. The results are very encouraging, suggesting further extension to more general frameworks.
Highlights
Compressed sensing (CS) [1, 2] is a novel signal acquisition technique that recovers an unknown signal from a small set of linear measurements
In Lasso techniques [6], a parameter λ related to k has to be chosen [7], whereas for greedy algorithms, such as orthogonal matching pursuit (OMP) [8] or compressive sampling matching pursuit (CoSaMP) [9], the performance and the number of iterations depend on k
We show that the performance of the proposed estimators of the sparsity degree depends on the Signal-to-noise ratio (SNR) = λ2k/σ 2 and that the traditional measure x 2/σ 2 has no significant effect in the estimation of the sparsity degree
Summary
Compressed sensing (CS) [1, 2] is a novel signal acquisition technique that recovers an unknown signal from a small set of linear measurements. In most of CS applications, it is usually assumed that an upper bound on the sparsity degree k is known before acquiring the signal. The optimal tuning of parameters requires the knowledge of the degree of sparsity of the signal. In Lasso techniques [6], a parameter λ related to k has to be chosen [7], whereas for greedy algorithms, such as orthogonal matching pursuit (OMP) [8] or compressive sampling matching pursuit (CoSaMP) [9], the performance and the number of iterations depend on k
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