Given an arbitrary matrix $A\in\mathbb{R}^{n\times n}$, we consider the fundamental problem of computing $Ax$ for any $x\in\mathbb{R}^n$ such that $Ax$ is $s$-sparse. While fast algorithms exist for particular choices of $A$, such as the discrete Fourier transform, there is currently no $o(n^2)$ algorithm that treats the unstructured case. In this paper, we devise a randomized approach to tackle the unstructured case. Our method relies on a representation of $A$ in terms of certain real-valued mutually unbiased bases derived from Kerdock sets. In the preprocessing phase of our algorithm, we compute this representation of $A$ in $O(n^3\log n)$ operations. Next, given any unit vector $x\in\mathbb{R}^n$ such that $Ax$ is $s$-sparse, our randomized fast transform uses this representation of $A$ to compute the entrywise $\epsilon$-hard threshold of $Ax$ with high probability in only $O(sn + \epsilon^{-2}\|A\|_{2\to\infty}^2n\log n)$ operations. In addition to a performance guarantee, we provide numerical results that demonstrate the plausibility of real-world implementation of our algorithm.
翻译:根据任意的 $A\ in\ mathbb{R\\\ ntime n$ 任意的矩阵 $A\ in\ mathb{R\\\\ n美元,我们考虑到计算美元美元的任何美元的基本问题。在计算美元的任何美元时,美元是美元,例如离散的Fourier变换,虽然存在对美元的特定选择的快速算法,但目前没有处理非结构化案件的$(n\\\\ 2美元)算法。在本文中,我们设计了一种随机化的处理非结构化案例的方法。我们的方法依赖于以来自Kerdock 的某种实际价值的相互公正基数计算$A$。在我们算法的预处理阶段,我们算出美元代表的美元代表是美元,而如果任何单位的矢量 $xin\ mathb{ { 美元处理非结构化案件,那么我们随机化的快速变换方式使用美元代表制的这个代表, 仅以美元= 美元\\\\\\\\\\ 美元 $Ax 美元高概率的运行。