# Manifold learning: what, how, and why

arXiv: https://arxiv.org/abs/2311.03757 · HTML: https://mlsanity.com/paper/2311.03757/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 108 references that are in the map.
- First posted on arXiv 7 Nov 2023.

## Builds on (most cited first, 20 of 20)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (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)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [Attraction-Repulsion Spectrum in Neighbor Embeddings](https://mlsanity.com/paper/2007.08902/) (2020)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations](https://mlsanity.com/paper/1902.05804/) (2019)
- [Intrinsic Dimension Estimation Using Wasserstein Distances](https://mlsanity.com/paper/2106.04018/) (2021)
- [Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs](https://mlsanity.com/paper/1910.13476/) (2019)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [Minimax Rates for Estimating the Dimension of a Manifold](https://mlsanity.com/paper/1605.01011/) (2016)
- [t-SNE, Forceful Colorings and Mean Field Limits](https://mlsanity.com/paper/2102.13009/) (2021)
- [Asymptotic theory for density ridges](https://mlsanity.com/paper/1406.5663/) (2014)
- [Stochastic Neighbor Embedding under f-divergences](https://mlsanity.com/paper/1811.01247/) (2018)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis](https://mlsanity.com/paper/1709.03615/) (2017)
- [Improved graph Laplacian via geometric self-consistency](https://mlsanity.com/paper/1406.0118/) (2014)
- [Manifold Learning via Manifold Deflation](https://mlsanity.com/paper/2007.03315/) (2020)

## Built on by (most cited first, 30 of 68)

- [Landing with the Score: Riemannian Optimization through Denoising](https://mlsanity.com/paper/2509.23357/) (2025)
- [Staying VIGILant: Mitigating Visual Laziness via Counterfactual Visual Alignment in MLLMs](https://mlsanity.com/paper/2606.26387/) (2026)
- [Towards aerodynamic surrogate modeling based on β-variational autoencoders](https://mlsanity.com/paper/2408.04969/) (2024)
- [DisRFM: Polar Riemannian Flow Matching for Structure-Preserving Graph Domain Adaptation](https://mlsanity.com/paper/2602.00656/) (2026)
- [Inductive Global and Local Manifold Approximation and Projection](https://mlsanity.com/paper/2406.08097/) (2024)
- [Adaptive Multi-Order Graph Regularized NMF with Dual Sparsity for Hyperspectral Unmixing](https://mlsanity.com/paper/2503.19258/) (2025)
- [GeoDM: Geometry-aware Distribution Matching for Dataset Distillation](https://mlsanity.com/paper/2512.08317/) (2025)
- [Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of Non-Euclidean Operators](https://mlsanity.com/paper/2404.10843/) (2024)
- [Sailing in high-dimensional spaces: Low-dimensional embeddings through angle preservation](https://mlsanity.com/paper/2406.09876/) (2024)
- [A review of unsupervised learning in astronomy](https://mlsanity.com/paper/2406.17316/) (2024)
- [Machine Learning of Slow Collective Variables and Enhanced Sampling via Spatial Techniques](https://mlsanity.com/paper/2412.20868/) (2024)
- [Adversarial Subspace Generation for Outlier Detection in High-Dimensional Data](https://mlsanity.com/paper/2504.07522/) (2025)
- [Cryo-em images are intrinsically low dimensional](https://mlsanity.com/paper/2504.11249/) (2025)
- [EmbedOR: Provable Cluster-Preserving Visualizations with Curvature-Based Stochastic Neighbor Embeddings](https://mlsanity.com/paper/2509.03703/) (2025)
- [Learning Reduced Representations for Quantum Classifiers](https://mlsanity.com/paper/2512.01509/) (2025)
- [The Semantic Bottleneck: Leveraging Semantic Representations for Non-Invasive Speech Decoding](https://mlsanity.com/paper/2609.10296/) (2026)
- [Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach](https://mlsanity.com/paper/2411.02058/) (2024)
- [Structure-preserving contrastive learning for spatial time series](https://mlsanity.com/paper/2502.06380/) (2025)
- [Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators](https://mlsanity.com/paper/2503.04649/) (2025)
- [Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularization](https://mlsanity.com/paper/2505.04412/) (2025)
- [Learning Design-Score Manifold to Guide Diffusion Models for Offline Optimization](https://mlsanity.com/paper/2506.05680/) (2025)
- [Learnable quantum spectral filters for hybrid graph neural networks](https://mlsanity.com/paper/2507.05640/) (2025)
- [Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems](https://mlsanity.com/paper/2507.07291/) (2025)
- [Beyond Motion Cues and Structural Sparsity: Revisiting Small Moving Target Detection](https://mlsanity.com/paper/2509.07654/) (2025)
- [Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning](https://mlsanity.com/paper/2510.17772/) (2025)
- [Inference for Similarity and Alignability between Noisy High-Dimensional Datasets](https://mlsanity.com/paper/2511.21074/) (2025)
- [Exploring Time Conditioning in Diffusion Generative Models from Disjoint Noisy Data Manifolds](https://mlsanity.com/paper/2604.25289/) (2026)
- [Datacube segmentation via Deep Spectral Clustering](https://mlsanity.com/paper/2401.17695/) (2024)
- [Strong denoising of financial time-series](https://mlsanity.com/paper/2408.05690/) (2024)
- [Pseudo-Nonlinear Data Augmentation: A Constrained Energy Minimization Viewpoint](https://mlsanity.com/paper/2410.00718/) (2024)

## Abstract (arXiv)

Manifold learning (ML), known also as non-linear dimension reduction, is a set of methods to find the low dimensional structure of data. Dimension reduction for large, high dimensional data is not merely a way to reduce the data; the new representations and descriptors obtained by ML reveal the geometric shape of high dimensional point clouds, and allow one to visualize, de-noise and interpret them. This survey presents the principles underlying ML, the representative methods, as well as their statistical foundations from a practicing statistician's perspective. It describes the trade-offs, and what theory tells us about the parameter and algorithmic choices we make in order to obtain reliable conclusions.