亚线性时间距离矩阵的低秩近似
Sublinear Time Low-Rank Approximation of Distance Matrices
AI总结：
本文研究了距离矩阵的低秩近似问题，证明在亚线性时间内可实现加法误差近似，并探讨了距离矩阵在亚线性时间内的相对误差近似限制。
摘要：
令P={p₁,p₂,…,pₙ}和Q={q₁,q₂,…,qₘ}为任意度量空间中的点集，令A为m×n的配对距离矩阵，其中A_{i,j}=d(p_i,q_j)。此类距离矩阵常用于软件包中，应用于学习图像流形、手写识别和多维展开等。为减少其描述大小，我们研究此类矩阵的低秩近似。我们的主要结果是证明对于任意底层距离度量d，可以在亚线性时间内实现加法误差低秩近似。我们注意到对于任意矩阵A，亚线性时间内实现此类保证是不可能的，因此我们的证明利用了距离矩阵的特殊性质。我们开发了一种基于加法投影成本保持采样的递归算法。然后证明，一般情况下，距离矩阵在亚线性时间内实现相对误差近似是不可能的，即使允许双标准解。此外，我们还证明，当P=Q且d是平方欧几里得距离时（而非度量，而是度量的平方），可以在亚线性时间内找到相对误差双标准解。
英文摘要：
Let $\mathbf{P}=\{ p_1, p_2, \ldots p_n \}$ and $\mathbf{Q} = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\mathbf{A}$ represent the $m\times n$ pairwise distance matrix with $\mathbf{A}_{i,j} = d(p_i, q_j)$. Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low-rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\mathbf{A}$, and consequently our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling. We then show that in general, relative error approximation in sublinear time is impossible for distance matrices, even if one allows for bicriteria solutions. Additionally, we show that if $\mathbf{P} = \mathbf{Q}$ and $d$ is the squared Euclidean distance, which is not a metric but rather the square of a metric, then a relative error bicriteria solution can be found in sublinear time.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
