General Perturbations of Sparse Signals in Compressed Sensing
Résumé
We analyze the Basis Pursuit recovery of signals when observing sparse data with general perturbations. Previous studies have only considered partially perturbed observations Ax+e. Here, x is a K-sparse signal which we wish to recover, A is a measurement matrix with more columns than rows, and e is simple additive noise. Our model also incorporates perturbations E (which result in multiplicative noise) to the matrix A in the form of (A + E)x + e: This completely perturbed framework extends the previous work of Candès, Romberg and Tao on stable signal recovery from incomplete and inaccurate measurements. Our results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level in the observation. Moreover, this accuracy is within a constant multiple of the best-case reconstruction using the technique of least squares.
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