Global Convergence of the Locally Competitive Algorithm (bibtex)
by , and
Abstract:
The Locally Competitive Algorithm (LCA) is a continuous-time dynamical system designed to solve the problem of sparse approximation. This class of approximation problems plays an important role in producing state-of-the-art results in many signal processing and inverse problems, and implementing the LCA in analog VLSI may significantly improve the time and power necessary to solve these optimization programs. The goal of this paper is to analyze the dynamical behavior of the LCA system and guarantee its convergence and stability. We show that the LCA converges from any initial point. We also show that fixed points of the system are extrema of the sparse approximation objective function when designed for a certain class of sparsity-inducing cost penalty. In addition, we prove that under certain conditions on the solution, the LCA converges in a finite number of switches (i.e., node threshold crossings).
Reference:
Global Convergence of the Locally Competitive AlgorithmA. Balavoine, C.J. Rozell and J.K. Romberg. In Proceedings of the IEEE Digital Signal Processing (DSP) Workshop, January 2011.
Bibtex Entry:
@InProceedings{balavoine.11,
  author = 	 {Balavoine, A. and Rozell, C.J. and Romberg, J.K.},
  title = 	 {Global Convergence of the {L}ocally {C}ompetitive {A}lgorithm},
  booktitle =	 {{Proceedings of the IEEE Digital Signal Processing (DSP) Workshop}},
  year =	 2011,
  month = {January},
  address =	 {Sedona, AZ},
abstract = {The Locally Competitive Algorithm (LCA) is a continuous-time dynamical system designed to solve the problem of sparse approximation. This class of approximation problems plays an important role in producing state-of-the-art results in many signal processing and inverse problems, and implementing the LCA in analog VLSI may significantly improve the time and power necessary to solve these optimization programs. The goal of this paper is to analyze the dynamical behavior of the LCA system and guarantee its convergence and stability. We show that the LCA converges from any initial point. We also show that fixed points of the system are extrema of the sparse approximation objective function when designed for a certain class of sparsity-inducing cost penalty. In addition, we prove that under certain conditions on the solution, the LCA converges in a finite number of switches (i.e., node threshold crossings).}
}
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