论文标题
围绕各个方程的线性方程系统的解决方案
Surrounding the solution of a Linear System of Equations from all sides
论文作者
论文摘要
假设$ a \ in \ mathbb {r}^{n \ times n} $是可逆的，我们正在寻找$ ax = b $的解决方案。给定初始猜测$ x_1 \ in \ mathbb {r} $，我们通过通过$ a $的行生成的超音台进行反思，我们可以生成一个无限序列$ $（x_k）_ {k = 1}^}^{\ infty} $，以使所有元素都能达到相同的距离。 = \ | x_1 -x \ | $。如果随机选择超平面，则序列上的平均值和$$ \ mathbb {e} \ left \ | x- \ frac {1} {m} \ sum_ {k = 1}^{m} {x_k} {x_k} \ right \ | \ leq \ frac {1 + \ | a \ | _f \ | a^{ - 1} \ |} {\ sqrt {m}} \ cdot \ cdot \ | x-x_1 \ |。这引入了一种纯粹的几何方式来攻击问题：是否有快速的方法来估算球体中心的位置，从了解球体上的许多点？我们的收敛率（与随机kaczmarz方法的汇合率）来自平均，一个人可以做得更好吗？
Suppose $A \in \mathbb{R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb{R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate an infinite sequence $(x_k)_{k=1}^{\infty}$ such that all elements have the same distance to the solution, i.e. $\|x_k - x\| = \|x_1 - x\|$. If the hyperplanes are chosen at random, averages over the sequence converge and $$ \mathbb{E} \left\| x - \frac{1}{m} \sum_{k=1}^{m}{ x_k} \right\| \leq \frac{1 + \|A\|_F \|A^{-1}\|}{\sqrt{m}} \cdot\|x-x_1\|.$$ The bound does not depend on the dimension of the matrix. This introduces a purely geometric way of attacking the problem: are there fast ways of estimating the location of the center of a sphere from knowing many points on the sphere? Our convergence rate (coinciding with that of the Random Kaczmarz method) comes from averaging, can one do better?
