  This paper studies the convergence rate of a continuous-time dynamical system
for L1-minimization, known as the Locally Competitive Algorithm (LCA). Solving
L1-minimization} problems efficiently and rapidly is of great interest to the
signal processing community, as these programs have been shown to recover
sparse solutions to underdetermined systems of linear equations and come with
strong performance guarantees. The LCA under study differs from the typical L1
solver in that it operates in continuous time: instead of being specified by
discrete iterations, it evolves according to a system of nonlinear ordinary
differential equations. The LCA is constructed from simple components, giving
it the potential to be implemented as a large-scale analog circuit.
  The goal of this paper is to give guarantees on the convergence time of the
LCA system. To do so, we analyze how the LCA evolves as it is recovering a
sparse signal from underdetermined measurements. We show that under appropriate
conditions on the measurement matrix and the problem parameters, the path the
LCA follows can be described as a sequence of linear differential equations,
each with a small number of active variables. This allows us to relate the
convergence time of the system to the restricted isometry constant of the
matrix. Interesting parallels to sparse-recovery digital solvers emerge from
this study. Our analysis covers both the noisy and noiseless settings and is
supported by simulation results.
