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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| Vydáno v: | Communications on pure and applied mathematics Ročník 66; číslo 8; s. 1241 - 1274 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.08.2013
John Wiley and Sons, Limited |
| Témata: | |
| ISSN: | 0010-3640, 1097-0312 |
| On-line přístup: | Získat plný text |
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| Abstract | 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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| AbstractList | 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 ...; that is, from data in which phase information is missing. We prove that if the vectors equation image 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 nlog 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-a-vis additive noise. (ProQuest: ... denotes formulae/symbols omitted.) 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. 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. |
| Author | Strohmer, Thomas Candès, Emmanuel J. Voroninski, Vladislav |
| Author_xml | – sequence: 1 givenname: Emmanuel J. surname: Candès fullname: Candès, Emmanuel J. email: candes@stanford.edu organization: Departments of Mathematics and of Statistics, Stanford University, Stanford CA 94305 – sequence: 2 givenname: Thomas surname: Strohmer fullname: Strohmer, Thomas email: strohmer@math.ucdavis.edu organization: Department of Mathematics, University of California at Davis, Davis CA 95616 – sequence: 3 givenname: Vladislav surname: Voroninski fullname: Voroninski, Vladislav email: vladv@math.berkeley.edu organization: Department of Mathematics, University of California at Berkeley, Berkeley CA 94720 |
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| CODEN | CPMAMV |
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| Title | PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming |
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