Low rank approximation of a matrix (hereafter LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with LRA at sublinear cost, that is, by using much fewer memory cells and flops than an input matrix has entries, but no sublinear cost algorithm can compute accurate LRA of the worst case input matrices or even of the matrices of small families in our Appendix. Nevertheless we prove that Cross-Approximation celebrated algorithms and even more primitive sublinear cost algorithms output quite accurate LRA for a large subclass of the class of all matrices that admit LRA and in a sense for most of such matrices. Moreover, we accentuate the power of sublinear cost LRA by means of multiplicative pre-processing of an input matrix, and this also reveals a link between C-A algorithms and Randomized and Sketching LRA algorithms. Our tests are in good accordance with our formal study.

翻译：矩阵的低秩近似（以下简称LRA）是数值线性与多重线性代数及数据挖掘与分析领域极为重要的研究方向。利用次线性代价（即使用远少于输入矩阵元素数量的存储单元和浮点运算次数）进行LRA计算是可行的，但次线性代价算法无法对最坏情况输入矩阵（甚至附录中展示的小规模矩阵族）计算出精确的LRA。尽管如此，我们证明：对于所有可进行LRA的矩阵大类中的子类，Cross-Approximation（交叉逼近）经典算法乃至更原始的次线性代价算法均能输出相当精确的LRA，且在某种意义上适用于此类矩阵中的绝大部分。此外，我们通过对输入矩阵进行乘法预处理来强调次线性代价LRA的效力，这同时揭示了交叉逼近算法与随机/草图化LRA算法之间的关联。我们的实验结果与形式化分析高度吻合。