从含噪几何图中谱恢复点云
Spectral Recovery of Point Clouds from Noisy Geometric Graphs
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中文总结 AI 辅助
本文研究从含噪高维数据生成的随机几何图中恢复低维潜在几何，提出谱嵌入算法，在谱间隙条件下利用邻接矩阵顶部特征向量和特征值近似恢复点云，并在嵌套球面和高维正弦曲线上验证。
中文摘要 AI 辅助
我们研究了从由含噪高维数据生成的随机几何图中恢复低维潜在几何的问题。具体而言，我们分析了在信号加噪声图模型上谱嵌入算法的性能，在该模型中，顶点与受高斯噪声扰动的点相关联，并且对于内积超过指定对齐阈值的点对，在它们之间包含边。在高维情形下，即点的数量$n$和环境维度$d$都趋于无穷大时，我们证明在谱间隙条件下，图的邻接矩阵的顶部特征向量和特征值可用于在正交变换意义下近似恢复点云。我们在从嵌套球面和高维正弦曲线采样的点云上展示了我们的结果。
英文摘要
We study the problem of recovering low-dimensional latent geometry from a random geometric graph generated by noisy, high-dimensional data. Specifically, we analyze the performance of a spectral embedding algorithm on the Signal+Noise Graph Model, in which vertices are associated to points perturbed by Gaussian noise, and edges are included for pairs whose inner product exceeds a specified alignment threshold. In the high-dimensional regime where the number $n$ of points and the ambient dimension $d$ both tend to infinity, we show that under a spectral gap condition, the top eigenvectors and eigenvalues of the graph's adjacency matrix can be used to approximately recover the point cloud up to an orthogonal transformation. We illustrate our results on point clouds sampled from nested spheres and high-dimensional sinusoid curves.
发表机构
- Duke University（杜克大学）
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