PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
Suppose we wish to recover a signal \input amssym $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} {\bi x} \in {\Bbb C}^n$ from m intensity measurements of the form $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} |\langle \bi x,\bi z_i \rangle|^2$, $i = 1, 2, \ldots, m$; that is, from data in which phase infor...
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| Veröffentlicht in: | Communications on pure and applied mathematics Jg. 66; H. 8; S. 1241 - 1274 |
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| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.08.2013
John Wiley and Sons, Limited |
| Schlagworte: | |
| ISSN: | 0010-3640, 1097-0312 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Suppose we wish to recover a signal
\input amssym $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} {\bi x} \in {\Bbb C}^n$
from m intensity measurements of the form $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} |\langle \bi x,\bi z_i \rangle|^2$, $i = 1, 2, \ldots, m$; that is, from data in which phase information is missing. We prove that if the vectors $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}}{\bi z}_i$
are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program–‐a trace‐norm minimization problem; this holds with large probability provided that m is on the order of $n {\log n}$, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis‐à‐vis additive noise. © 2012 Wiley Periodicals, Inc. |
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| Bibliographie: | istex:5328C24230599923C4F226FA5D1CA5619F6267F6 ark:/67375/WNG-7G9HC512-8 ArticleID:CPA21432 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 0010-3640 1097-0312 |
| DOI: | 10.1002/cpa.21432 |