magnipy: Metric Space Magnitude Computations 🔍

This is a repository for computing the magnitude of a metric space, which encodes the effective size, diversity, and geometry of a metric space. Given a dataset or distance matrix, magnitude measures the effective number of distinct points in the space at a scale of dissimilarity between observations. We introduce the following codebase to compute and compare the magnitude of metric spaces.

Main Functionalities and Classes

Magnipy: For in-depth magnitude computations on a single metric space.

The functionalities of Magnipy for an individual metric space include:

• Computing the metric space's distance matrix
• Calculating the similarity matrix from the distances
• Executing an automated scale-finding procedure to find suitable evaluation scales
• Computing the magnitude weight of each point across multiple distance scales
• Evaluating and plotting magnitude functions across varying distance scales
• Estimating magnitude dimension profiles and calculating the magnitude dimension to quantify intrinsic dimensionality

Diversipy: For comparing magnitude (and thus diversity) across multiple datasets.

The functionalities of Diversipy for a list of spaces (that share the same distance metric) include:

• Executing an automated scale-finding procedure and the determining a common evaluation interval across datasets
• Computing magnitude functions across varying distance scales
• Calculating MagArea, the area under a magnitude function, a multi-scale measure of the intrinsic diversity of a dataset
• Calculating MagDiff, the area between magnitude functions, to measure the difference in diversity between datasets

Dependencies

To get started,

1. Clone this repository locally into a directory of your choosing
2. Create and activate a new Python virtual environment

Our dependencies are managed using the poetry package manager. Using your activated virtual environment, run the following to install poetry:

$ pip install poetry

With poetry installed, run the following command from the main directory to download the necessary dependencies:

$ poetry install

Tutorials

Tutorials demonstrating the main functionalities can be found under the notebooks folder.

The following tutorials demonstrate how to initialize and utiltize the corresponding classes:

• magnipy_tutorial.ipynb: Using Magnipy for computing the magnitude of one metric space
• diversipy_tutorial.ipynb: Using Diversipy for comparing the magnitude of multiple metric spaces

The following supplementary demos are also provided:

• mode_dropping.ipynb: Using MagDiff for detecting mode dropping / mode collapse

Citation

Please consider citing our work!

@inproceedings{limbeck2024metric,
  title         = {Metric Space Magnitude for Evaluating the Diversity of Latent Representations}, 
  author        = {Katharina Limbeck and Rayna Andreeva and Rik Sarkar and Bastian Rieck},
  booktitle     = {Advances in Neural Information Processing Systems},
  volume        = {37},
  year          = {2024}
}

@inproceedings{andreeva2023metric,
  title         = {Metric Space Magnitude and Generalisation in Neural Networks},
  author        = {Andreeva, Rayna and Limbeck, Katharina and Rieck, Bastian and Sarkar, Rik},
  year          = {2023},
  booktitle     = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning~(TAG-ML)},
  volume        = {221},
  pages         = {242--253}
}