Ordinary Differential Equation-based Sparse Signal Recovery
Ordinary Differential Equation-based Sparse Signal Recovery
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DOI：
10.48550/arxiv.2303.16431
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发表时间：
2023-03
期刊：
影响因子：
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通讯作者：
T. Wadayama;Ayano Nakai-Kasai
中科院分区：
文献类型：
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作者：
T. Wadayama;Ayano Nakai-Kasai
这项研究研究了使用连续时间动力学系统来稀疏信号恢复。所提出的动力系统的形式是从套索目标函数的梯度流得出的非线性普通微分方程（ODE）的形式。使用Euler方法的数值模拟证明了这种基于ODE的方法的稀疏信号恢复过程。连续时间动力系统的状态最终会收敛到对应于目标函数的最小值的平衡点。为了深入了解系统的局部收敛属性，应用了平衡点周围的线性近似，从而产生封闭形式的误差演化ode。该分析显示了收敛到平衡点的行为。此外，提出了一个变分优化问题，以优化时间依赖的正则化参数，以提高收敛速度和解决方案质量。引入了深层展开的变化优化方法作为解决此优化问题的一种手段，并通过数值实验验证其有效性。
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.