# FloDR: An invertible dimensionality reduction method based on a normalising flow

arXiv: https://arxiv.org/abs/2607.26278 · HTML: https://mlsanity.com/paper/2607.26278/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 28 Jul 2026.

## Builds on (most cited first, 21 of 21)

- [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)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (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)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Parametric UMAP embeddings for representation and semi-supervised learning](https://mlsanity.com/paper/2009.12981/) (2020)
- [Learning Ordered Representations with Nested Dropout](https://mlsanity.com/paper/1402.0915/) (2014)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Attraction-Repulsion Spectrum in Neighbor Embeddings](https://mlsanity.com/paper/2007.08902/) (2020)
- [Minimum-Distortion Embedding](https://mlsanity.com/paper/2103.02559/) (2021)
- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Dimension Reduction with Locally Adjusted Graphs](https://mlsanity.com/paper/2412.15426/) (2024)
- [Ordering Dimensions with Nested Dropout Normalizing Flows](https://mlsanity.com/paper/2006.08777/) (2020)
- [Nonlinear Isometric Manifold Learning for Injective Normalizing Flows](https://mlsanity.com/paper/2203.03934/) (2022)
- [Invertible Manifold Learning for Dimension Reduction](https://mlsanity.com/paper/2010.04012/) (2020)
- [SQuadMDS: a lean Stochastic Quartet MDS improving global structure preservation in neighbor embedding like t-SNE and UMAP](https://mlsanity.com/paper/2202.12087/) (2022)
- [ShaRP: Shape-Regularized Multidimensional Projections](https://mlsanity.com/paper/2306.00554/) (2023)
- [Evaluating Autoencoders for Parametric and Invertible Multidimensional Projections](https://mlsanity.com/paper/2504.16831/) (2025)

## Built on by (most cited first, 1 of 1)

- [NFMap: Topographic 2D Manifold Learning with Invertible Normalizing Flows](https://mlsanity.com/paper/2610.16750/) (2026)

## Abstract (arXiv)

It is common for two-dimensional embeddings of high-dimensional data to be read far beyond what they can support. Distances in and between clusters, the meaning behind empty spaces, and the amount of structure hidden at each point are generally invisible in the output of methods such as t-SNE and UMAP. This is because the information that could support the meaning of these properties is discarded during the optimisation process. Here, we present FloDR, a dimensionality reduction method that embeds data through an invertible normalising flow. While FloDR only uses the first two output coordinates to create a two-dimensional embedding, it retains the remaining coordinates rather than discarding them. In addition to the embedding, an exact inverse and an exact density are properties of a trained mapping, which enable diagnostic visualisations that are computed from the exact inverse of the model that drew the layout rather than from an approximate one. Specifically, we draw two fields, the conditional spread, which measures how much of the original data remains undetermined at each embedding position in input units, and the hidden contrast, which measures how much information about a labelled contrast the two plotted coordinates discard. Both fields are rendered with a prespecified test against a held out portion of the input data and a bootstrap confidence. A field that fails the test is reported as refused.