A flexible convex relaxation for phase retrieval
| Autor: | Sohail Bahmani, Justin Romberg |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Statistics and Probability
Semidefinite programming Vapnik-Chervonenkis theory Logarithm convex relaxation 010102 general mathematics Estimator 020206 networking & telecommunications 02 engineering and technology 01 natural sciences 90C25 Compressed sensing Quadratic equation Vapnik–Chervonenkis theory 0202 electrical engineering electronic engineering information engineering 63P30 0101 mathematics Statistics Probability and Uncertainty Extreme point Phase retrieval Algorithm 62F10 Mathematics |
| Zdroj: | Electron. J. Statist. 11, no. 2 (2017), 5254-5281 |
| Popis: | We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with “lifting” and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric “slab”. Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector. We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. We demonstrate the effectiveness of the proposed method through numerical simulations using both independent random measurements and coded diffraction patterns. ¶ We also extend this formulation to include sparsity constraints on the target vector. With this additional constraint, we show that, considering “nested” measurements, the number of phaseless measurements needed to recover the sparse vector is essentially the same (to within a logarithmic factor) as the number of linear measurements needed by standard compressive sensing techniques. |
| Databáze: | OpenAIRE |
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