Phaseless compressive sensing using partial support information
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Phaseless compressive sensing using partial support information Zhiyong Zhou1 · Jun Yu2 Received: 17 December 2018 / Accepted: 10 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study the recovery conditions of weighted 1 minimization for real-valued signal reconstruction from phaseless compressive sensing measurements when partial support information is available. A strong restricted isometry property condition is provided to ensure the stable recovery. Moreover, we present the weighted null space property as the sufficient and necessary condition for the success of k-sparse phaseless recovery via weighted 1 minimization. Numerical experiments are conducted to illustrate our results. Keywords Phaseless compressive sensing · Partial support information · Strong restricted isometry property · Weighted null space property
1 Introduction Compressive sensing aims to recover an unknown signal from the underdetermined linear measurements (see [8,9] for a comprehensive view). It is known as phase retrieval or phaseless compressive sensing when there is no phase information. The phaseless compressive sensing problem has recently attracted considerable research interests and many algorithms have been proposed to solve this problem. Existing literature includes [2–4,7,12,16,18], to name a few. Specifically, the goal of phaseless compressive sensing is to recover x ∈ R N up to a unimodular scaling constant from noisy magnitude measurements y = |Ax| + e ∈ Rm with the measurement matrix A = (a1 , . . . , am )T ∈ Rm×N , |Ax| = (|a1 , x|, . . . , |am , x|)T and the noise term e ∈ Rm . When x is sparse or compressible, the stable recovery can be guaranteed by solving the following 1 minimization problem
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Zhiyong Zhou [email protected] Jun Yu [email protected]
1
Department of Statistics, Zhejiang University City College, Hangzhou 310015, China
2
Department of Mathematics and Mathematical Statistics, Umeå University, 901 87 Umeå, Sweden
123
Z. Zhou, J. Yu
min z1 subject to |Az| − y2 ≤ ε,
z∈R N
(1)
provided that the measurement matrix A satisfies the strong restricted isometry property (SRIP) [11,19]. In the noiseless case, the first sufficient and necessary condition was presented in [20] by proposing a new version of null space property for the phase retrieval problem. In this paper, we generalize the existing theoretical framework for phaseless compressive sensing to incorporate partial support information, where we consider the case that an estimate of the support of the signal is available. We follow the similar notations and arguments in [10,22]. For an arbitrary signal x ∈ R N , let x k be its best k-term approximation, so that x k minimizes x − f 1 over all k-sparse vectors f . Let T0 be the support of x k , where T0 ⊂ {1, . . . , N } and |T0 | ≤ k. Let T˜ , the support estimate, be a subset of {1, 2 . . . , N } with cardinality |T˜ | = ρk, where ρ ≥ 0 and |T˜ ∩ T0 | = αρk with 0 ≤ α ≤ 1. Here the parameter ρ determines the ratio of the size of the estimat
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