# Impact of signal-to-noise ratio and bandwidth on graph Laplacian spectrum from high-dimensional noisy point cloud

arXiv: https://arxiv.org/abs/2011.10725 · HTML: https://mlsanity.com/paper/2011.10725/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2020.

## Built on by (most cited first, 14 of 14)

- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
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- [How do kernel-based sensor fusion algorithms behave under high dimensional noise?](https://mlsanity.com/paper/2111.10940/) (2021)
- [Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets](https://mlsanity.com/paper/2202.01671/) (2022)
- [Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators](https://mlsanity.com/paper/2405.12317/) (2024)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Augmentation Invariant Manifold Learning](https://mlsanity.com/paper/2211.00460/) (2022)
- [Design a Metric Robust to Complicated High Dimensional Noise for Efficient Manifold Denoising](https://mlsanity.com/paper/2401.03921/) (2024)
- [Boundary Detection Algorithm Inspired by Locally Linear Embedding](https://mlsanity.com/paper/2406.18456/) (2024)
- [Conditional Multidimensional Scaling with Incomplete Conditioning Data](https://mlsanity.com/paper/2509.16627/) (2025)
- [Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization](https://mlsanity.com/paper/2510.02308/) (2025)
- [Generalized Robust Adaptive-Bandwidth Multi-View Manifold Learning in High Dimensions with Noise](https://mlsanity.com/paper/2602.10530/) (2026)

## Abstract (arXiv)

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.