在流式奇异值分解（SVD）中，可能无限矩阵的$d$维行作为$\mathbb{R}^d$中的点顺序到达。一个$ε$ -coreset是一个（小得多的）矩阵，它的行到任何超平面的平方距离之和近似于原始矩阵到$1 \pm ε$因子的平方距离之和。我们的主要结果是，我们可以保持一个$ε$ -coreset，而只存储$O(d \log^2 d / ε^2)$行。已知的$Ω(d / ε^2)$行的下限表明这几乎是最优的。此外，我们的coreset的每一行都是输入行的加权子集。这是非常可取的，因为它：（1）保持稀疏性;（2）易于解释;（3）避免精度误差;（4）适用于对输入有约束的问题。以前的SVD流结果返回输入的子集，需要存储$Ω(d \log^3 n / ε^2)$行，其中$n$是到目前为止看到的行数。我们的算法，存储独立的$n$，是第一个结果，使用有限的内存上无限的流。我们支持我们的研究结果与实验的维基百科数据集基准对国家的最先进的算法。
英文摘要：
In streaming Singular Value Decomposition (SVD), $d$-dimensional rows of a possibly infinite matrix arrive sequentially as points in $\mathbb{R}^d$. An $ε$-coreset is a (much smaller) matrix whose sum of square distances of the rows to any hyperplane approximates that of the original matrix to a $1 \pm ε$ factor. Our main result is that we can maintain a $ε$-coreset while storing only $O(d \log^2 d / ε^2)$ rows. Known lower bounds of $Ω(d / ε^2)$ rows show that this is nearly optimal. Moreover, each row of our coreset is a weighted subset of the input rows. This is highly desirable since it: (1) preserves sparsity; (2) is easily interpretable; (3) avoids precision errors; (4) applies to problems with constraints on the input. Previous streaming results for SVD that return a subset of the input required storing $Ω(d \log^3 n / ε^2)$ rows where $n$ is the number of rows seen so far. Our algorithm, with storage independent of $n$, is the first result that uses finite memory on infinite streams. We support our findings with experiments on the Wikipedia dataset benchmarked against state-of-the-art algorithms.
Streaming algorithms for computing coresets and $k$-median clustering in the Hamming space
CNRS & DIENS, École normale supérieure de Paris, PSL Research University（法国国家科学研究中心与信息与电子工程研究所，巴黎高等师范学院，巴黎文理研究大学）；Department of Mathematics and Computer Science (IMADA), University of Southern Denmark（南丹麦大学数学与计算机科学系（IMADA））
Turnstile $\ell_p$ leverage score sampling with applications
Dortmund Data Science Center（多特蒙德数据科学中心）；Faculties of Statistics and Computer Science, TU Dortmund University（多特蒙德工业大学统计与计算机科学学院）；Lamarr-Institute for Machine Learning and Artificial Intelligence, TU Dortmund University（多特蒙德工业大学拉马尔机器学习与人工智能研究所）
