基于低秩近似的投影聚类核心集快速构造
Faster Coreset Construction for Projective Clustering via Low-Rank Approximation
摘要：
在这项工作中，我们提出了一个随机coreset建设的投影聚类，其中包括计算一组$k$最接近的$j$维线性（仿射）子空间的一组给定的$d$维度的$n$向量。令$A \in \mathbb{R}^{n\times d}$为输入矩阵。Feldman \textit{et. al.}的早期确定性coreset构造依赖于计算$A$的SVD。用于SVD的最佳已知算法需要$\min\{nd^2, n^2d\}$时间，这对于$n$和$d$的大值可能是不可行的。通过将矩阵$A$的行投影到与矩阵$A$的右奇异向量非常接近的正交向量上，提出了一种核集构造方法.因此，当$k$和$j$的值较小时，与Feldman \textit{et. al.}的算法相比，我们能够实现更快的算法，同时保持几乎相同的近似。我们还受益于空间以及利用输入数据集的稀疏性。我们的方法的另一个优点是，它可以在流设置中非常有效地构建。
英文摘要：
In this work, we present a randomized coreset construction for projective clustering, which involves computing a set of $k$ closest $j$-dimensional linear (affine) subspaces of a given set of $n$ vectors in $d$ dimensions. Let $A \in \mathbb{R}^{n\times d}$ be an input matrix. An earlier deterministic coreset construction of Feldman \textit{et. al.} relied on computing the SVD of $A$. The best known algorithms for SVD require $\min\{nd^2, n^2d\}$ time, which may not be feasible for large values of $n$ and $d$. We present a coreset construction by projecting the rows of matrix $A$ on some orthonormal vectors that closely approximate the right singular vectors of $A$. As a consequence, when the values of $k$ and $j$ are small, we are able to achieve a faster algorithm, as compared to the algorithm of Feldman \textit{et. al.}, while maintaining almost the same approximation. We also benefit in terms of space as well as exploit the sparsity of the input dataset. Another advantage of our approach is that it can be constructed in a streaming setting quite efficiently.
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