# Inductive Global and Local Manifold Approximation and Projection

arXiv: https://arxiv.org/abs/2406.08097 · HTML: https://mlsanity.com/paper/2406.08097/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 47 references that are in the map.
- First posted on arXiv 12 Jun 2024.

## Builds on (most cited first, 14 of 14)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [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)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Parametric UMAP embeddings for representation and semi-supervised learning](https://mlsanity.com/paper/2009.12981/) (2020)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Extendable and invertible manifold learning with geometry regularized autoencoders](https://mlsanity.com/paper/2007.07142/) (2020)
- [DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling](https://mlsanity.com/paper/1711.06011/) (2017)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Geodesic Distance Estimation with Spherelets](https://mlsanity.com/paper/1907.00296/) (2019)
- [Dimensionality reduction to maximize prediction generalization capability](https://mlsanity.com/paper/2003.00470/) (2020)
- [Minimax Estimation of Distances on a Surface and Minimax Manifold Learning in the Isometric-to-Convex Setting](https://mlsanity.com/paper/2011.12478/) (2020)

## Built on by (most cited first, 3 of 3)

- [Assessing and improving reliability of neighbor embedding methods: a map-continuity perspective](https://mlsanity.com/paper/2410.16608/) (2024)
- [Harnessing Data Asymmetry: Manifold Learning in the Finsler World](https://mlsanity.com/paper/2603.11396/) (2026)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)

## Abstract (arXiv)

Nonlinear dimensional reduction with the manifold assumption, often called manifold learning, has proven its usefulness in a wide range of high-dimensional data analysis. The significant impact of t-SNE and UMAP has catalyzed intense research interest, seeking further innovations toward visualizing not only the local but also the global structure information of the data. Moreover, there have been consistent efforts toward generalizable dimensional reduction that handles unseen data. In this paper, we first propose GLoMAP, a novel manifold learning method for dimensional reduction and high-dimensional data visualization. GLoMAP preserves locally and globally meaningful distance estimates and displays a progression from global to local formation during the course of optimization. Furthermore, we extend GLoMAP to its inductive version, iGLoMAP, which utilizes a deep neural network to map data to its lower-dimensional representation. This allows iGLoMAP to provide lower-dimensional embeddings for unseen points without needing to re-train the algorithm. iGLoMAP is also well-suited for mini-batch learning, enabling large-scale, accelerated gradient calculations. We have successfully applied both GLoMAP and iGLoMAP to the simulated and real-data settings, with competitive experiments against the state-of-the-art methods.