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Greedy subspace pursuit for joint sparse recovery. (English) Zbl 07042028
Summary: In the joint sparse recovery, where the objective is to recover a signal matrix \(X_0\) of size \(n \times l\) or a set \(\Omega\) of its nonzero row indices from incomplete measurements, subspace-based greedy algorithms improving MUSIC such as subspace-augmented MUSIC and sequential compressive MUSIC have been proposed to improve the reconstruction performance of \(X_0\) and \(\Omega\) with a computational efficiency even when \(\mathrm{rank}(X_0) \leq k := | \Omega |\). However, the main limitation of the MUSIC-like methods is that they most likely fail to recover the signal when a partial support estimate of \(k - \mathrm{rank}(X_0)\) indices for their input is not fully correct. We proposed a computationally efficient algorithm called two-stage iterative method to detect the remained support (T-IDRS), its special version termed by two-stage orthogonal subspace matching pursuit (TSMP), and its variant called TSMP with sparse Bayesian learning (TSML) by exploiting more than the sparsity \(k\) to estimate the signal matrix. They improve on the MUSIC-like methods such that these are guaranteed to recover the signal and its support while the existing MUSIC-like methods will fail in the practically significant case of MMV when \(\mathrm{rank}(X_0) / k\) is sufficiently small. Numerical simulations demonstrate that the proposed schemes have low complexities and most likely outperform other related methods. A condition of the minimum \(m\) required for TSMP to recover the signal matrix is derived in the noiseless case to be applicable to a wide class of the sensing matrix.
MSC:
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
65F22 Ill-posedness and regularization problems in numerical linear algebra
Software:
SPGL1
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