# Invertible Manifold Learning for Dimension Reduction

arXiv: https://arxiv.org/abs/2010.04012 · HTML: https://mlsanity.com/paper/2010.04012/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2020.

## Built on by (most cited first, 5 of 5)

- [GenURL: A General Framework for Unsupervised Representation Learning](https://mlsanity.com/paper/2110.14553/) (2021)
- [Generative Invertible Quantum Neural Networks](https://mlsanity.com/paper/2302.12906/) (2023)
- [Local Distance Preserving Auto-encoders using Continuous k-Nearest Neighbours Graphs](https://mlsanity.com/paper/2206.05909/) (2022)
- [FloDR: An invertible dimensionality reduction method based on a normalising flow](https://mlsanity.com/paper/2607.26278/) (2026)
- [Deep Invertible Autoencoders for Dimensionality Reduction of Dynamical Systems](https://mlsanity.com/paper/2603.13496/) (2026)

## Abstract (arXiv)

Dimension reduction (DR) aims to learn low-dimensional representations of high-dimensional data with the preservation of essential information. In the context of manifold learning, we define that the representation after information-lossless DR preserves the topological and geometric properties of data manifolds formally, and propose a novel two-stage DR method, called invertible manifold learning (inv-ML) to bridge the gap between theoretical information-lossless and practical DR. The first stage includes a homeomorphic sparse coordinate transformation to learn low-dimensional representations without destroying topology and a local isometry constraint to preserve local geometry. In the second stage, a linear compression is implemented for the trade-off between the target dimension and the incurred information loss in excessive DR scenarios. Experiments are conducted on seven datasets with a neural network implementation of inv-ML, called i-ML-Enc. Empirically, i-ML-Enc achieves invertible DR in comparison with typical existing methods as well as reveals the characteristics of the learned manifolds. Through latent space interpolation on real-world datasets, we find that the reliability of tangent space approximated by the local neighborhood is the key to the success of manifold-based DR algorithms.