We propose superfast (aka sublinear cost) algorithms for (i) computation and (ii) iterative refinement of a Low Rank Approximation (LRA) of a matrix. A superfast algorithm only accesses a small fraction of all entries of a matrix and for a worst-case input cannot output its close LRA or even verify whether a fixed candidate matrix is indeed a close LRA. Nevertheless, we first recall some well-known algorithms that solve these problems in linear or super-linear time and then propose their superfast modifications, which promise to succeed for a large class of real-world matrices according to our analysis and numerical tests.

翻译：我们提出用于矩阵低秩近似（LRA）计算及其迭代改进的超快速（即次线性代价）算法。超快速算法仅访问矩阵全部条目中的一小部分，对于最坏情况下的输入，无法输出其精确低秩近似，甚至无法验证固定候选矩阵是否为精确低秩近似。尽管如此，我们首先回顾若干在线性时间或超线性时间内解决这些问题的知名算法，随后提出其超快速改进版本。根据我们的分析与数值测试，这些改进算法有望在大量实际矩阵上取得成功。