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Minimum-Distortion Embedding

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PyPI version Conda Version

The official documentation for PyMDE is available at

This repository accompanies the monograph Minimum-Distortion Embedding.

PyMDE is a Python library for computing vector embeddings for finite sets of items, such as images, biological cells, nodes in a network, or any other abstract object.

What sets PyMDE apart from other embedding libraries is that it provides a simple but general framework for embedding, called Minimum-Distortion Embedding (MDE). With MDE, it is easy to recreate well-known embeddings and to create new ones, tailored to your particular application.

PyMDE is competitive in runtime with more specialized embedding methods. With a GPU, it can be even faster.


PyMDE can be enjoyed by beginners and experts alike. It can be used to:

  • visualize datasets, small or large;
  • generate feature vectors for supervised learning;
  • compress high-dimensional vector data;
  • draw graphs (in up to orders of magnitude less time than packages like NetworkX);
  • create custom embeddings, with custom objective functions and constraints (such as having uncorrelated feature columns);
  • and more.

PyMDE is very young software, under active development. If you run into issues, or have any feedback, please reach out by filing a Github issue.

This README gives a very brief overview of PyMDE. Make sure to read the official documentation at, which has in-depth tutorials and API documentation.


PyMDE is available on the Python Package Index, and on Conda Forge.

To install with pip, use

pip install pymde

Alternatively, to install with conda, use

conda install -c pytorch -c conda-forge pymde

PyMDE has the following requirements:

  • Python >= 3.7
  • numpy >= 1.17.5
  • scipy
  • torch >= 1.7.1
  • torchvision >= 0.8.2
  • pynndescent
  • requests

Getting started

Getting started with PyMDE is easy. For embeddings that work out-of-the box, we provide two main functions:


which preserves the local structure of original data, and


which preserves pairwise distances or dissimilarity scores in the original data.

Arguments. The input to these functions is the original data, represented either as a data matrix in which each row is a feature vector, or as a (possibly sparse) graph encoding pairwise distances. The embedding dimension is specified by the embedding_dim keyword argument, which is 2 by default.

Return value. The return value is an MDE object. Calling the embed() method on this object returns an embedding, which is a matrix (torch.Tensor) in which each row is an embedding vector. For example, if the original input is a data matrix of shape (n_items, n_features), then the embedding matrix has shape (n_items, embeddimg_dim).

We give examples of using these functions below.

Preserving neighbors

The following code produces an embedding of the MNIST dataset (images of handwritten digits), in a fashion similar to LargeVis, t-SNE, UMAP, and other neighborhood-based embeddings. The original data is a matrix of shape (70000, 784), with each row representing an image.

import pymde

mnist = pymde.datasets.MNIST()
embedding = pymde.preserve_neighbors(, verbose=True).embed()
pymde.plot(embedding, color_by=mnist.attributes['digits'])

Unlike most other embedding methods, PyMDE can compute embeddings that satisfy constraints. For example:

embedding = pymde.preserve_neighbors(, constraint=pymde.Standardized(), verbose=True).embed()
pymde.plot(embedding, color_by=mnist.attributes['digits'])

The standardization constraint enforces the embedding vectors to be centered and have uncorrelated features.

Preserving distances

The function pymde.preserve_distances is useful when you're more interested in preserving the gross global structure instead of local structure.

Here's an example that produces an embedding of an academic coauthorship network, from Google Scholar. The original data is a sparse graph on roughly 40,000 authors, with an edge between authors who have collaborated on at least one paper.

import pymde

google_scholar = pymde.datasets.google_scholar()
embedding = pymde.preserve_distances(, verbose=True).embed()
pymde.plot(embedding, color_by=google_scholar.attributes['coauthors'], color_map='viridis', background_color='black')

More collaborative authors are colored brighter, and are near the center of the embedding.

Example notebooks

We have several example notebooks that show how to use PyMDE on real (and synthetic) datasets.


To cite our work, please use the following BibTex entry.

  author  = {Agrawal, Akshay and Ali, Alnur and Boyd, Stephen},
  title   = {Minimum-Distortion Embedding},
  journal = {arXiv},
  year    = {2021},

PyMDE was designed and developed by Akshay Agrawal.

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