线性算子的稀疏表示与矩阵乘积的逼近
On sparse representations of linear operators and the approximation of matrix products
摘要：
到目前为止，稀疏表示已被利用，主要是在背景下的鲁棒估计功能在一个嘈杂的环境中，从几个测量。在这种情况下，所考虑的信号类稀疏的基础的存在被用来减少必要的测量的数量，同时控制近似误差。在本文中，我们反而专注于数值分析中的应用，通过线性算子的稀疏表示，目标是最小化在这些算子上执行基本运算（这里是乘法）所需的运算数量。我们表示一个线性算子的秩一运算符的总和，并显示如何稀疏表示，保证低近似误差的产品可以从分析诱导二次型。这种结构反过来又产生新的算法计算近似矩阵产品。
英文摘要：
Thus far, sparse representations have been exploited largely in the context of robustly estimating functions in a noisy environment from a few measurements. In this context, the existence of a basis in which the signal class under consideration is sparse is used to decrease the number of necessary measurements while controlling the approximation error. In this paper, we instead focus on applications in numerical analysis, by way of sparse representations of linear operators with the objective of minimizing the number of operations needed to perform basic operations (here, multiplication) on these operators. We represent a linear operator by a sum of rank-one operators, and show how a sparse representation that guarantees a low approximation error for the product can be obtained from analyzing an induced quadratic form. This construction in turn yields new algorithms for computing approximate matrix products.
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