# Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach

arXiv: https://arxiv.org/abs/2203.00126 · HTML: https://mlsanity.com/paper/2203.00126/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 107 references that are in the map.
- First posted on arXiv 28 Feb 2022.

## Builds on (most cited first, 18 of 18)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data](https://mlsanity.com/paper/2105.07536/) (2021)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [An Analysis of the t-SNE Algorithm for Data Visualization](https://mlsanity.com/paper/1803.01768/) (2018)
- [Optimality of Spectral Clustering in the Gaussian Mixture Model](https://mlsanity.com/paper/1911.00538/) (2019)
- [Spectral algorithms for tensor completion](https://mlsanity.com/paper/1612.07866/) (2016)
- [Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation](https://mlsanity.com/paper/2006.02037/) (2020)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Analysis of spectral clustering algorithms for community detection: the general bipartite setting](https://mlsanity.com/paper/1803.04547/) (2018)
- [Spectral Convergence of the connection Laplacian from random samples](https://mlsanity.com/paper/1306.1587/) (2013)
- [Impact of signal-to-noise ratio and bandwidth on graph Laplacian spectrum from high-dimensional noisy point cloud](https://mlsanity.com/paper/2011.10725/) (2020)
- [Graph Based Gaussian Processes on Restricted Domains](https://mlsanity.com/paper/2010.07242/) (2020)
- [How do kernel-based sensor fusion algorithms behave under high dimensional noise?](https://mlsanity.com/paper/2111.10940/) (2021)
- [Improving Spectral Clustering using the Asymptotic Value of the Normalised Cut](https://mlsanity.com/paper/1703.09975/) (2017)
- [Connection graph Laplacian methods can be made robust to noise](https://mlsanity.com/paper/1405.6231/) (2014)

## Built on by (most cited first, 8 of 8)

- [A Spectral Method for Assessing and Combining Multiple Data Visualizations](https://mlsanity.com/paper/2210.13711/) (2022)
- [Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators](https://mlsanity.com/paper/2405.12317/) (2024)
- [Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets](https://mlsanity.com/paper/2407.01718/) (2024)
- [Morephy-Net: An Evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed Neural Operator Learning Networks](https://mlsanity.com/paper/2509.00663/) (2025)
- [Inference for Similarity and Alignability between Noisy High-Dimensional Datasets](https://mlsanity.com/paper/2511.21074/) (2025)
- [IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation](https://mlsanity.com/paper/2505.06288/) (2025)
- [Conditional Multidimensional Scaling with Incomplete Conditioning Data](https://mlsanity.com/paper/2509.16627/) (2025)
- [Generalized Robust Adaptive-Bandwidth Multi-View Manifold Learning in High Dimensions with Noise](https://mlsanity.com/paper/2602.10530/) (2026)

## Abstract (arXiv)

We propose a kernel-spectral embedding algorithm for learning low-dimensional nonlinear structures from high-dimensional and noisy observations, where the datasets are assumed to be sampled from an intrinsically low-dimensional manifold and corrupted by high-dimensional noise. The algorithm employs an adaptive bandwidth selection procedure which does not rely on prior knowledge of the underlying manifold. The obtained low-dimensional embeddings can be further utilized for downstream purposes such as data visualization, clustering and prediction. Our method is theoretically justified and practically interpretable. Specifically, we establish the convergence of the final embeddings to their noiseless counterparts when the dimension and size of the samples are comparably large, and characterize the effect of the signal-to-noise ratio on the rate of convergence and phase transition. We also prove convergence of the embeddings to the eigenfunctions of an integral operator defined by the kernel map of some reproducing kernel Hilbert space capturing the underlying nonlinear structures. Numerical simulations and analysis of three real datasets show the superior empirical performance of the proposed method, compared to many existing methods, on learning various manifolds in diverse applications.