Convergence and Rate Analysis of Neural Networks for Sparse Approximation

We present an analysis of the Locally Competitive Algorithm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation problems (e.g., approximating a vector from a dictionary using just a few nonzero coefficients). This class of problems plays a significant role in...

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Veröffentlicht in:	IEEE transaction on neural networks and learning systems Jg. 23; H. 9; S. 1377 - 1389
Hauptverfasser:	Balavoine, A., Romberg, J., Rozell, C. J.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York, NY IEEE 01.09.2012 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:	Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Artificial intelligence Computer science; control theory; systems Computer Simulation Connectionism. Neural networks Convergence Dictionaries Exact sciences and technology Exponential convergence global stability Least squares approximation locally competitive algorithm Lyapunov function Lyapunov methods Models, Statistical Neural networks Neural Networks (Computer) nonsmooth objective Optimization Pattern Recognition, Automated - methods Sample Size sparse approximation Studies Theoretical computing Hopfield neural nets Exponential convergence Dictionaries Competitiveness Global optimum global stability locally competitive algorithm Network management Neural network Function approximation Competitive algorithms Convergence speed nonsmooth objective, sparse approximation Convergence rate Signal processing Sparse representation Objective function Numerical convergence Hopfield model Lyapunov function
ISSN:	2162-237X, 2162-2388, 2162-2388
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Abstract	We present an analysis of the Locally Competitive Algorithm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation problems (e.g., approximating a vector from a dictionary using just a few nonzero coefficients). This class of problems plays a significant role in both theories of neural coding and applications in signal processing. However, the LCA lacks analysis of its convergence properties, and previous results on neural networks for nonsmooth optimization do not apply to the specifics of the LCA architecture. We show that the LCA has desirable convergence properties, such as stability and global convergence to the optimum of the objective function when it is unique. Under some mild conditions, the support of the solution is also proven to be reached in finite time. Furthermore, some restrictions on the problem specifics allow us to characterize the convergence rate of the system by showing that the LCA converges exponentially fast with an analytically bounded convergence rate. We support our analysis with several illustrative simulations.
AbstractList	We present an analysis of the Locally Competitive Algorithm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation problems (e.g., approximating a vector from a dictionary using just a few nonzero coefficients). This class of problems plays a significant role in both theories of neural coding and applications in signal processing. However, the LCA lacks analysis of its convergence properties, and previous results on neural networks for nonsmooth optimization do not apply to the specifics of the LCA architecture. We show that the LCA has desirable convergence properties, such as stability and global convergence to the optimum of the objective function when it is unique. Under some mild conditions, the support of the solution is also proven to be reached in finite time. Furthermore, some restrictions on the problem specifics allow us to characterize the convergence rate of the system by showing that the LCA converges exponentially fast with an analytically bounded convergence rate. We support our analysis with several illustrative simulations.We present an analysis of the Locally Competitive Algorithm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation problems (e.g., approximating a vector from a dictionary using just a few nonzero coefficients). This class of problems plays a significant role in both theories of neural coding and applications in signal processing. However, the LCA lacks analysis of its convergence properties, and previous results on neural networks for nonsmooth optimization do not apply to the specifics of the LCA architecture. We show that the LCA has desirable convergence properties, such as stability and global convergence to the optimum of the objective function when it is unique. Under some mild conditions, the support of the solution is also proven to be reached in finite time. Furthermore, some restrictions on the problem specifics allow us to characterize the convergence rate of the system by showing that the LCA converges exponentially fast with an analytically bounded convergence rate. We support our analysis with several illustrative simulations. We present an analysis of the Locally Competitive Algotihm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation problems (e.g., approximating a vector from a dictionary using just a few nonzero coefficients). This class of problems plays a significant role in both theories of neural coding and applications in signal processing. However, the LCA lacks analysis of its convergence properties, and previous results on neural networks for nonsmooth optimization do not apply to the specifics of the LCA architecture. We show that the LCA has desirable convergence properties, such as stability and global convergence to the optimum of the objective function when it is unique. Under some mild conditions, the support of the solution is also proven to be reached in finite time. Furthermore, some restrictions on the problem specifics allow us to characterize the convergence rate of the system by showing that the LCA converges exponentially fast with an analytically bounded convergence rate. We support our analysis with several illustrative simulations. We present an analysis of the Locally Competitive Algorithm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation problems (e.g., approximating a vector from a dictionary using just a few nonzero coefficients). This class of problems plays a significant role in both theories of neural coding and applications in signal processing. However, the LCA lacks analysis of its convergence properties, and previous results on neural networks for nonsmooth optimization do not apply to the specifics of the LCA architecture. We show that the LCA has desirable convergence properties, such as stability and global convergence to the optimum of the objective function when it is unique. Under some mild conditions, the support of the solution is also proven to be reached in finite time. Furthermore, some restrictions on the problem specifics allow us to characterize the convergence rate of the system by showing that the LCA converges exponentially fast with an analytically bounded convergence rate. We support our analysis with several illustrative simulations.
Author	Romberg, J. Rozell, C. J. Balavoine, A.
Author_xml	– sequence: 1 givenname: A. surname: Balavoine fullname: Balavoine, A. email: aurele.balavoine@gatech.edu organization: Sch. of Electr. & Comput. Eng., Georgia Inst. of Technol., Atlanta, GA, USA – sequence: 2 givenname: J. surname: Romberg fullname: Romberg, J. email: jrom@ece.gatech.edu organization: Sch. of Electr. & Comput. Eng., Georgia Inst. of Technol., Atlanta, GA, USA – sequence: 3 givenname: C. J. surname: Rozell fullname: Rozell, C. J. email: crozell@gatech.edu organization: Sch. of Electr. & Comput. Eng., Georgia Inst. of Technol., Atlanta, GA, USA
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Keywords	Hopfield neural nets Exponential convergence Dictionaries Competitiveness Global optimum global stability locally competitive algorithm Network management Neural network Function approximation Competitive algorithms Convergence speed nonsmooth objective, sparse approximation Convergence rate Signal processing Sparse representation Objective function Numerical convergence Hopfield model Lyapunov function
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Snippet	We present an analysis of the Locally Competitive Algorithm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation... We present an analysis of the Locally Competitive Algotihm (LCA), which is a Hopfield-style neural network that efficiently solves sparse approximation...
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SubjectTerms	Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Artificial intelligence Computer science; control theory; systems Computer Simulation Connectionism. Neural networks Convergence Dictionaries Exact sciences and technology Exponential convergence global stability Least squares approximation locally competitive algorithm Lyapunov function Lyapunov methods Models, Statistical Neural networks Neural Networks (Computer) nonsmooth objective Optimization Pattern Recognition, Automated - methods Sample Size sparse approximation Studies Theoretical computing
Title	Convergence and Rate Analysis of Neural Networks for Sparse Approximation
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