信噪比和带宽对高维噪声点云图拉普拉斯谱的影响
Impact of signal-to-noise ratio and bandwidth on graph Laplacian spectrum from high-dimensional noisy point cloud
摘要：
本文系统地研究了在非零条件下，由高维噪声随机点云构造的基于核的图拉普拉斯算子（GL）的谱。该问题的动机是通过研究模型时，干净的信号是从一个流形，嵌入在一个低维欧几里德子空间，并损坏高维噪声采样。我们量化了信号和噪声如何在不同的信噪比（SNR）区域相互作用，并报告了由此产生的GL特有的光谱行为。此外，我们探讨了所选择的内核带宽对GL在不同SNR区域的频谱的影响，这导致内核带宽的自适应选择与实际数据中的常见做法相一致。这一结果为从理论上理解实践者在数据集有噪声时如何应用GL铺平了道路。
英文摘要：
We systematically study the spectrum of kernel-based graph Laplacian (GL) constructed from high-dimensional and noisy random point cloud in the nonnull setup. The problem is motived by studying the model when the clean signal is sampled from a manifold that is embedded in a low-dimensional Euclidean subspace, and corrupted by high-dimensional noise. We quantify how the signal and noise interact over different regions of signal-to-noise ratio (SNR), and report the resulting peculiar spectral behavior of GL. In addition, we explore the impact of chosen kernel bandwidth on the spectrum of GL over different regions of SNR, which lead to an adaptive choice of kernel bandwidth that coincides with the common practice in real data. This result paves the way to a theoretical understanding of how practitioners apply GL when the dataset is noisy.
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