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.

翻译：本研究探讨了利用连续时间动力系统进行稀疏信号恢复。所提出的动力系统采用非线性常微分方程的形式，该方程源自Lasso目标函数的梯度流。通过使用欧拉法的数值模拟，展示了基于常微分方程的稀疏信号恢复过程。连续时间动力系统的状态最终收敛到目标函数最小值对应的平衡点。为了深入理解系统的局部收敛特性，在平衡点附近进行了线性近似，得到了闭式误差演化常微分方程。该分析揭示了向平衡点收敛的行为。此外，提出了一种变分优化问题来优化时变正则化参数，以同时提升收敛速度和解的质量。引入了深度展开变分优化方法作为求解该优化问题的手段，并通过数值实验验证了其有效性。