论文标题
线性逆问题的放松正则化
Relaxed regularization for linear inverse problems
论文作者
论文摘要
我们考虑了$ \ min_ {x} \ frac {1} {2} {2} \ vert ax -b \ vert_2^2 + \ m athcal {r}（lx）$的正规化最小二乘问题。最近，Zheng等人，2019年，提出了一种称为“稀疏正规化回归（SR3）的算法”，该算法通过引入辅助变量$ y $来采用分裂策略，并求解$ \ min_ {x，y}，y}，y} \ frac {1} {1} {2} {2} {2} {2} \ vert ax -b \ b \ vert_2_2_2^2 \ frac { y \ vert_2^2 + \ Mathcal {r}（x）$。通过最小化变量$ x $，我们获得了等效的系统$ \ min_ {y} \ frac {1} {2} {2} \ vertf_κY -g_κ\ vert_2^2+\ m athcal {r}（r}（y）$。在我们的工作中，我们将SR3方法视为近似解决正规化问题的一种方法。我们总体上分析了放松问题的条件，并为$κ$的$f_κ$提供了SVD的表达。 此外，我们将原始问题的帕累托曲线与放松的问题联系起来，并用$κ$量化了放松所产生的误差。最后，我们提出了一种有效的迭代方法，用于解决不精确的内部迭代问题。数值示例说明了方法。
We consider regularized least-squares problems of the form $\min_{x} \frac{1}{2}\Vert Ax - b\Vert_2^2 + \mathcal{R}(Lx)$. Recently, Zheng et al., 2019, proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable $y$ and solves $\min_{x,y} \frac{1}{2}\Vert Ax - b\Vert_2^2 + \fracκ{2}\Vert Lx - y\Vert_2^2 + \mathcal{R}(x)$. By minimizing out the variable $x$ we obtain an equivalent system $\min_{y} \frac{1}{2} \Vert F_κy - g_κ\Vert_2^2+\mathcal{R}(y)$. In our work we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of $F_κ$ as a function of $κ$. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem and we quantify the error incurred by relaxation in terms of $κ$. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach.
