Title:
Mathematical analysis of a dynamical system for sparse recovery

dc.contributor.advisor Romberg, Justin
dc.contributor.author Balavoine, Aurele
dc.contributor.committeeMember Rozell, Christopher J.
dc.contributor.committeeMember Hasler, Jennifer 0.
dc.contributor.committeeMember Yezzi, Anthony J.
dc.contributor.committeeMember Balcan, Maria-Florina
dc.contributor.committeeMember Davenport, Mark A.
dc.contributor.department Electrical and Computer Engineering
dc.date.accessioned 2014-05-22T15:33:14Z
dc.date.available 2014-05-22T15:33:14Z
dc.date.created 2014-05
dc.date.issued 2014-04-04
dc.date.submitted May 2014
dc.date.updated 2014-05-22T15:33:14Z
dc.description.abstract This thesis presents the mathematical analysis of a continuous-times system for sparse signal recovery. Sparse recovery arises in Compressed Sensing (CS), where signals of large dimension must be recovered from a small number of linear measurements, and can be accomplished by solving a complex optimization program. While many solvers have been proposed and analyzed to solve such programs in digital, their high complexity currently prevents their use in real-time applications. On the contrary, a continuous-time neural network implemented in analog VLSI could lead to significant gains in both time and power consumption. The contributions of this thesis are threefold. First, convergence results for neural networks that solve a large class of nonsmooth optimization programs are presented. These results extend previous analysis by allowing the interconnection matrix to be singular and the activation function to have many constant regions and grow unbounded. The exponential convergence rate of the networks is demonstrated and an analytic expression for the convergence speed is given. Second, these results are specialized to the L1-minimization problem, which is the most famous approach to solving the sparse recovery problem. The analysis relies on standard techniques in CS and proves that the network takes an efficient path toward the solution for parameters that match results obtained for digital solvers. Third, the convergence rate and accuracy of both the continuous-time system and its discrete-time equivalent are derived in the case where the underlying sparse signal is time-varying and the measurements are streaming. Such a study is of great interest for practical applications that need to operate in real-time, when the data are streaming at high rates or the computational resources are limited. As a conclusion, while existing analysis was concentrated on discrete-time algorithms for the recovery of static signals, this thesis provides convergence rate and accuracy results for the recovery of static signals using a continuous-time solver, and for the recovery of time-varying signals with both a discrete-time and a continuous-time solver.
dc.description.degree Ph.D.
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/51882
dc.language.iso en_US
dc.publisher Georgia Institute of Technology
dc.subject Sparse recovery
dc.subject Neural network
dc.subject L1-minimization
dc.subject Nonsmooth optimization
dc.subject Compressed sensing
dc.subject Tracking
dc.subject ISTA
dc.subject LCA
dc.subject.lcsh Sparse matrices
dc.subject.lcsh Signal processing Digital techniques
dc.subject.lcsh Mathematical optimization
dc.title Mathematical analysis of a dynamical system for sparse recovery
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Romberg, Justin
local.contributor.corporatename School of Electrical and Computer Engineering
local.contributor.corporatename College of Engineering
relation.isAdvisorOfPublication 23ff0d70-23a6-4f87-bde3-5f3427d03dfe
relation.isOrgUnitOfPublication 5b7adef2-447c-4270-b9fc-846bd76f80f2
relation.isOrgUnitOfPublication 7c022d60-21d5-497c-b552-95e489a06569
thesis.degree.level Doctoral
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