快速最小二乘逼近
Faster Least Squares Approximation
摘要：
最小二乘近似是一种技术，用于找到没有精确解的线性方程组的近似解。在典型的设置中，让$n$为约束的数量，$d$为变量的数量，其中$n \gg d$。然后，现有的精确方法在$O(nd^2)$时间内找到解向量。我们提出了两个随机算法，提供非常准确的相对误差近似的最佳值和最小二乘近似问题的解向量更快地比现有的精确算法。我们的两个算法预处理的数据与随机阿达玛变换。然后，一个均匀随机采样约束，并解决这些约束上的较小问题，另一个执行稀疏随机投影，并解决这些投影坐标上的较小问题。在这两种情况下，解决较小的问题提供了相对误差近似，并且如果$n$充分大于$d$，则近似解可以在$O(nd \log d)$时间内计算。
英文摘要：
Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. In a typical setting, one lets $n$ be the number of constraints and $d$ be the number of variables, with $n \gg d$. Then, existing exact methods find a solution vector in $O(nd^2)$ time. We present two randomized algorithms that provide very accurate relative-error approximations to the optimal value and the solution vector of a least squares approximation problem more rapidly than existing exact algorithms. Both of our algorithms preprocess the data with the Randomized Hadamard Transform. One then uniformly randomly samples constraints and solves the smaller problem on those constraints, and the other performs a sparse random projection and solves the smaller problem on those projected coordinates. In both cases, solving the smaller problem provides relative-error approximations, and, if $n$ is sufficiently larger than $d$, the approximate solution can be computed in $O(nd \log d)$ time.
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