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dc.contributor.authorRush, Cynthiaen
dc.contributor.authorVenkataramanan, Ramjien
dc.date.accessioned2018-05-29T12:11:11Z
dc.date.available2018-05-29T12:11:11Z
dc.date.issued2016en
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/276254
dc.description.abstractApproximate message passing (AMP) refers to a class of efficient algorithms for statistical estimation in high-dimensional problems such as compressed sensing and low-rank matrix estimation. This paper analyzes the performance of AMP in the regime where the problem dimension is large but finite. For concreteness, we consider the setting of high-dimensional regression, where the goal is to estimate a high-dimensional vector $\beta_0$ from a noisy measurement $y=A \beta_0 + w$. AMP is a low-complexity, scalable algorithm for this problem. Under suitable assumptions on the measurement matrix $A$, AMP has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution. Previous proofs of the validity of state evolution have all been asymptotic convergence results. In this paper, we derive a concentration inequality for AMP with i.i.d.\ Gaussian measurement matrices with finite size $n \times N$. The result shows that the probability of deviation from the state evolution prediction falls exponentially in $n$. This provides theoretical support for empirical findings that have demonstrated excellent agreement of AMP performance with state evolution predictions for moderately large dimensions. The concentration inequality also indicates that the number of AMP iterations $t$ can grow no faster than order $\frac{\log n}{\log \log n}$ for the performance to be close to the state evolution predictions with high probability. The analysis can be extended to obtain similar non-asymptotic results for AMP in other settings such as low-rank matrix estimation.
dc.description.sponsorshipMarie Curie Career Integration Grant under Grant Agreement Number 631489.
dc.publisherIEEE
dc.titleFinite-sample analysis of Approximate Message Passing.en
dc.typeArticle
prism.endingPage759
prism.publicationDate2016en
prism.publicationNameISITen
prism.startingPage755
dc.identifier.doi10.17863/CAM.23535
dcterms.dateAccepted2018-03-06en
rioxxterms.versionAM*
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserveden
rioxxterms.licenseref.startdate2016en
dc.contributor.orcidVenkataramanan, Ramji [0000-0001-7915-5432]
rioxxterms.typeJournal Article/Reviewen
pubs.funder-project-idEuropean Commission (631489)
dc.identifier.urlhttp://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7532279en
rioxxterms.freetoread.startdate2017-08-17


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