双随机归一化图拉普拉斯算子：收敛于流形拉普拉斯算子及对离群噪声的鲁棒性
Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise
AI总结：
本文证明双随机归一化图拉普拉斯算子收敛于流形拉普拉斯算子，给出速率，并证明其对离群噪声的鲁棒性，提出近似归一化方法保持一致性。
摘要：
双随机归一化提供了基于图的数据分析中图拉普拉斯算子的另一种归一化方式，并可通过Sinkhorn-Knopp（SK）迭代高效计算。本文证明了当$n$个数据点从嵌入在可能高维空间中的一般$d$维流形上独立同分布采样时，双随机归一化图拉普拉斯算子以一定速率收敛于流形（加权）拉普拉斯算子。在$n \to \infty$和核带宽$\epsilon \to 0$的特定联合极限下，图拉普拉斯算子（在2-范数下）的点态收敛速率被证明为$O(n^{-1/(d/2+3)})$（在有限大的$n$下，忽略对数因子），该速率在$\epsilon \sim n^{-1/(d/2+3)}$的缩放条件下实现。当流形数据被离群噪声破坏时，我们从理论上证明了图拉普拉斯算子的点态一致性，其速率与干净流形数据的速率相匹配，并加上一个额外项，该项与噪声向量之间以及噪声向量与数据向量的内积的有界性成正比。我们的分析表明，并非精确的双随机归一化，而是近似的双随机归一化能达到相同的一致性速率，受此启发，我们提出了一个近似且受约束的矩阵缩放问题，可通过提前终止的SK迭代求解。数值实验支持我们的理论结果，并展示了双随机归一化图拉普拉斯算子对高维离群噪声的鲁棒性。
英文摘要：
Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates, when $n$ data points are i.i.d. sampled from a general $d$-dimensional manifold embedded in a possibly high-dimensional space. Under certain joint limit of $n \to \infty$ and kernel bandwidth $ε\to 0$, the point-wise convergence rate of the graph Laplacian operator (under 2-norm) is proved to be $ O( n^{-1/(d/2+3)})$ at finite large $n$ up to log factors, achieved at the scaling of $ε\sim n^{-1/(d/2+3)} $. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency which matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors among themselves and with data vectors. Motivated by our analysis, which suggests that not exact bi-stochastic normalization but an approximate one will achieve the same consistency rate, we propose an approximate and constrained matrix scaling problem that can be solved by SK iterations with early termination. Numerical experiments support our theoretical results and show the robustness of bi-stochastically normalized graph Laplacian to high-dimensional outlier noise.
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