Spectral Recovery of Point Clouds from Noisy Geometric Graphs
Abstract
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.
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BibTeX
@misc{brailovskaya2026spectral,
title = {{Spectral Recovery of Point Clouds from Noisy Geometric Graphs}},
author = {Tatiana Brailovskaya and Nicholas A. Cook and Sofia Poinelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08634},
eprint = {2610.08634},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08634},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Spectral Recovery of Point Clouds from Noisy Geometric Graphs
AU - Tatiana Brailovskaya
AU - Nicholas A. Cook
AU - Sofia Poinelli
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08634
UR - https://arxiv.org/abs/2610.08634
AB - 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.
ER -