{"title":"无小球概率假设的相位恢复:相位提升的恢复保证","authors":"F. Krahmer, Yi-Kai Liu","doi":"10.1109/SAMPTA.2015.7148966","DOIUrl":null,"url":null,"abstract":"We study the problem of recovering an unknown vector x ε R<sup>n</sup> from measurements of the form y<sub>i</sub> = |a<sup>T</sup><sub>i</sub> x|<sup>2</sup> (for i = 1,..., m), where the vectors a<sub>i</sub> ε R<sup>n</sup> are chosen independently at random, with each coordinate a<sub>ij</sub> ε R being chosen independently from a fixed sub-Gaussian distribution D. However, without making additional assumptions on the random variables a<sub>ij</sub> - for example on the behavior of their small ball probabilities - it may happen some vectors x cannot be uniquely recovered. We show that for any sub-Gaussian distribution V, with no additional assumptions, it is still possible to recover most vectors x. More precisely, one can recover those vectors x that are not too peaky in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The recovery guarantees in this paper are for the PhaseLift algorithm, a tractable convex program based on a matrix formulation of the problem. We prove uniform recovery of all not too peaky vectors from m = 0(n) measurements, in the presence of noise. This extends previous work on PhaseLift by Candès and Li [8].","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"34 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Phase retrieval without small-ball probability assumptions: Recovery guarantees for phaselift\",\"authors\":\"F. Krahmer, Yi-Kai Liu\",\"doi\":\"10.1109/SAMPTA.2015.7148966\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the problem of recovering an unknown vector x ε R<sup>n</sup> from measurements of the form y<sub>i</sub> = |a<sup>T</sup><sub>i</sub> x|<sup>2</sup> (for i = 1,..., m), where the vectors a<sub>i</sub> ε R<sup>n</sup> are chosen independently at random, with each coordinate a<sub>ij</sub> ε R being chosen independently from a fixed sub-Gaussian distribution D. However, without making additional assumptions on the random variables a<sub>ij</sub> - for example on the behavior of their small ball probabilities - it may happen some vectors x cannot be uniquely recovered. We show that for any sub-Gaussian distribution V, with no additional assumptions, it is still possible to recover most vectors x. More precisely, one can recover those vectors x that are not too peaky in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The recovery guarantees in this paper are for the PhaseLift algorithm, a tractable convex program based on a matrix formulation of the problem. We prove uniform recovery of all not too peaky vectors from m = 0(n) measurements, in the presence of noise. This extends previous work on PhaseLift by Candès and Li [8].\",\"PeriodicalId\":311830,\"journal\":{\"name\":\"2015 International Conference on Sampling Theory and Applications (SampTA)\",\"volume\":\"34 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2015-05-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2015 International Conference on Sampling Theory and Applications (SampTA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/SAMPTA.2015.7148966\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2015 International Conference on Sampling Theory and Applications (SampTA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/SAMPTA.2015.7148966","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Phase retrieval without small-ball probability assumptions: Recovery guarantees for phaselift
We study the problem of recovering an unknown vector x ε Rn from measurements of the form yi = |aTi x|2 (for i = 1,..., m), where the vectors ai ε Rn are chosen independently at random, with each coordinate aij ε R being chosen independently from a fixed sub-Gaussian distribution D. However, without making additional assumptions on the random variables aij - for example on the behavior of their small ball probabilities - it may happen some vectors x cannot be uniquely recovered. We show that for any sub-Gaussian distribution V, with no additional assumptions, it is still possible to recover most vectors x. More precisely, one can recover those vectors x that are not too peaky in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The recovery guarantees in this paper are for the PhaseLift algorithm, a tractable convex program based on a matrix formulation of the problem. We prove uniform recovery of all not too peaky vectors from m = 0(n) measurements, in the presence of noise. This extends previous work on PhaseLift by Candès and Li [8].