This study investigates the use of continuous-time dynamical systems for
sparse signal recovery. The proposed dynamical system is in the form of a
nonlinear ordinary differential equation (ODE) derived from the gradient flow
of the Lasso objective function. The sparse signal recovery process of this
ODE-based approach is demonstrated by numerical simulations using the Euler
method. The state of the continuous-time dynamical system eventually converges
to the equilibrium point corresponding to the minimum of the objective
function. To gain insight into the local convergence properties of the system,
a linear approximation around the equilibrium point is applied, yielding a
closed-form error evolution ODE. This analysis shows the behavior of
convergence to the equilibrium point. In addition, a variational optimization
problem is proposed to optimize a time-dependent regularization parameter in
order to improve both convergence speed and solution quality. The deep
unfolded-variational optimization method is introduced as a means of solving
this optimization problem, and its effectiveness is validated through numerical
experiments.Comment: 13 page