# PyMDE: Minimum-Distortion Embedding¶

PyMDE is a Python library for computing vector embeddings of items, such as
images, biological cells, nodes in a network, or any other type of abstract
object. The embeddings are designed to minimally distort relationships between
pairs of items, while possibly satisfying some constraints.
PyMDE is based on the monograph *Minimum-Distortion Embedding*,
which introduced a simple but general framework for embedding.

You can use PyMDE to:

visualize datasets, small or large;

embed into arbitrarily large dimensions (much greater than 2 or 3);

generate feature vectors for supervised learning;

compress high-dimensional vector data;

draw graphs;

and more.

Unlike other popular embedding methods, PyMDE also lets you

compute large embeddings quickly, using a GPU;

fit embeddings that satisfy constraints, such as having uncorrelated feature columns;

design custom embeddings (and recreate well-known ones, like PCA, Laplacian eigenmap, LargeVis, and UMAP);

sanity-check embeddings using principled methods;

look for outliers in original data.

## Example¶

Here’s an example that shows how to use PyMDE to visualize high-dimensional vectors (which in this case are images from the MNIST dataset).

```
import pymde
mnist = pymde.datasets.MNIST()
embedding = pymde.preserve_neighbors(mnist.data, embedding_dim=2, verbose=True).embed()
pymde.plot(embedding, color_by=mnist.attributes['digits'])
```

Each point represents an image in the dataset, and every point is colored by the digit represented by its underlying image. However, only the raw pixel data was used to create the embedding. This example embedded into two dimensions, allowing us to plot the result, but PyMDE supports embedding into much larger dimensions.

This is just one of the many things that PyMDE can do.

## Why use PyMDE?¶

There are many reasons to try out PyMDE. Here are some.

**PyMDE is expressive.**
PyMDE is based on a simple but general framework for embedding, called
*Minimum-Distortion Embedding* (MDE). The MDE framework generalizes well-known
methods like PCA, spectral embedding, multi-dimensional scaling, LargeVis, and
UMAP. With PyMDE, it is easy to recreate well-known embeddings and to create
new ones, tailored to your particular application.

Unlike other embedding methods, PyMDE lets you impose
constraints on the embedding. For example, the below GIFs show PyMDE computing
three different kinds of embeddings of MNIST. The first and third
embeddings are constrained to be *standardized* (which roughly means they have
uncorrelated features), while the second is unconstrained. Standardized
embeddings are favorably scaled for downstream machine learning tasks, such
as supervised learning.

**PyMDE is easy to use.**
It’s easy to get started with PyMDE. With just one line of code, you can
embed high-dimensional vectors or the nodes of a graph into any target
dimension. Creating custom embeddings is also
easy, because PyMDE separates the description of an embedding from how it is
computed: you say what kind of embedding you want, and PyMDE will compute it
for you.

In addition to computing embeddings, PyMDE can plot them (if the embedding dimension is three or less), create GIFs of the embedding process, help you compare different embeddings, preprocess your data, and more.

**PyMDE is principled.**
The MDE framework provides principled methods to debug and analyze
embeddings. These methods can be used to make your embeddings more faithful
to your original data, and to search for anomalies and outliers in the original
data.

**PyMDE is fast.**
PyMDE is competitive in runtime with more specialized embedding methods, and
can scale to datasets with millions of items. With a GPU, it can be even
faster. PyMDE scales well with the embedding dimension, meaning that you can
easily embed into dimensions such as 50, 100, or 250.