论文标题
边缘反射方法比交替预测更好
The circumcentered-reflection method achieves better rates than alternating projections
论文作者
论文摘要
我们研究了绕线反射方法（CRM）的收敛速率，以解决凸的可行性问题，并将其与交替投影方法（MAP）进行比较。在一个错误结合的假设下，我们证明两种方法都根据误差的参数进行线性收敛，并以渐近常数收敛，并且为CRM派生的一种方法严格比MAP的误差。接下来，我们分析两类相当通用的示例。在第一个中，凸组之间的角度接近交点附近的零，因此地图序列会收敛，但是CRM仍然享有线性收敛。在第二类示例中，集合之间的角度不会消失，MAP表现出其标准行为，即线性收敛，但也许令人惊讶的是，CRM达到了超线性收敛。
We study the convergence rate of the Circumcentered-Reflection Method (CRM) for solving the convex feasibility problem and compare it with the Method of Alternating Projections (MAP). Under an error bound assumption, we prove that both methods converge linearly, with asymptotic constants depending on a parameter of the error bound, and that the one derived for CRM is strictly better than the one for MAP. Next, we analyze two classes of fairly generic examples. In the first one, the angle between the convex sets approaches zero near the intersection, so that the MAP sequence converges sublinearly, but CRM still enjoys linear convergence. In the second class of examples, the angle between the sets does not vanish and MAP exhibits its standard behavior, i.e., it converges linearly, yet, perhaps surprisingly, CRM attains superlinear convergence.
