Manify 🪐

Manify is a Python library for non-Euclidean representation learning. It is built on top of geoopt and follows scikit-learn API conventions. The library supports a variety of workflows involving (products of) Riemannian manifolds, including:

• All basic manifold operations (e.g. exponential map, logarithmic map, parallel transport, and distance computations)
• Sampling Gaussian distributions and Gaussian mixtures
• Learning embeddings of data on product manifolds, using features and/or distances
• Training machine learning models on manifold-valued embeddings, including decision trees, random forests, SVMs, perceptrons, and neural networks.
• Clustering manifold-valued data using Riemannian fuzzy K-Means

Installation

There are two ways to install manify:

1. From PyPI:

   pip install manify

2. From GitHub (recommended due to active development of the repo):

   pip install git+https://github.com/pchlenski/manify

Quick Example

import manify
from manify.utils.dataloaders import load_hf
from sklearn.model_selection import train_test_split

# Load Polblogs graph from HuggingFace
features, dists, adj, labels = load_hf("polblogs")

# Create an S^4 x H^4 product manifold
pm = manify.ProductManifold(signature=[(1.0, 4), (-1.0, 4)])

# Learn embeddings (Gu et al (2018) method)
embedder = manify.CoordinateLearning(pm=pm)
X_embedded = embedder.fit_transform(X=None, D=dists, burn_in_iterations=200, training_iterations=800)

# Train and evaluate classifier (Chlenski et al (2025) method)
X_train, X_test, y_train, y_test = train_test_split(X_embedded, labels)
model = manify.ProductSpaceDT(pm=pm, max_depth=3, task="classification")
model.fit(X_train, y_train)
print(model.score(X_test, y_test))

Tutorial

The official tutorial for Manify can be found in tutorial.ipynb. This Jupyter notebook contains a comprehensive overview of all of the library's core features.

Modules

Manifold Operations

• manify.manifolds - Tools for generating Riemannian manifolds and product manifolds

Curvature Estimation

• manify.curvature_estimation.delta_hyperbolicity - Compute delta-hyperbolicity of a metric space
• manify.curvature_estimation.greedy_method - Greedy selection of near-optimal signatures
• manify.curvature_estimation.sectional_curvature - Sectional curvature estimation using Toponogov's theorem

Embedders

• manify.embedders.coordinate_learning - Coordinate learning and optimization
• manify.embedders.siamese - Siamese network embedder
• manify.embedders.vae - Product space variational autoencoder

Predictors

• manify.predictors.nn - Neural network layers
• manify.predictors.decision_tree - Decision tree and random forest predictors
• manify.predictors.kappa_gcn - Kappa GCN
• manify.predictors.perceptron - Product space perceptron
• manify.predictors.svm - Product space SVM

Clustering

• manify.clustering.fuzzy_kmeans - Riemannian fuzzy K-Means for clustering

Optimizers

• manify.optimizers.radan - Riemannian version of Adan optimizer

Utilities

• manify.utils.benchmarks - Tools for benchmarking
• manify.utils.dataloaders - Loading datasets
• manify.utils.link_prediction - Preprocessing graphs with link prediction
• manify.utils.visualization - Tools for visualization

Archival branches

This repo has a number of archival branches that contain code from previous versions of the library when it was under active development. These branches are not maintained and are provided for reference only:

• Dataset-Generation. This branch contains code used to generate the datasets found in manify.utils.dataloaders.
• notebook-archive. This branch contains dozens of Jupyter notebooks and datasets that were used to develop the library and carry out various benchmarks for the Mixed Curvature Decision Trees and Random Forests paper.

Contributing

Please read our contributing guide for details on how to contribute to the project.

References

If you use our work, please cite the Manify paper:

@misc{chlenski2025manifypythonlibrarylearning,
      title={Manify: A Python Library for Learning Non-Euclidean Representations}, 
      author={Philippe Chlenski and Kaizhu Du and Dylan Satow and Itsik Pe'er},
      year={2025},
      eprint={2503.09576},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2503.09576}, 
}

Additionally, if you use one of the methods implemented in manify, please cite the original papers:

• CoordinateLearning: Gu et al. "Learning Mixed-Curvature Representations in Product Spaces." ICLR 2019. https://openreview.net/forum?id=HJxeWnCcF7
• ProductSpaceVAE: Skopek et al. "Mixed-Curvature Variational Autoencoders." ICLR 2020. https://openreview.net/forum?id=S1g6xeSKDS
• SiameseNetwork: Based on Siamese networks: Chopra et al. "Learning a Similarity Metric Discriminatively, with Application to Face Verification." CVPR 2005. https://ieeexplore.ieee.org/document/1467314
• ProductSpaceDT and ProductSpaceRF: Chlenski et al. "Mixed Curvature Decision Trees and Random Forests." ICML 2025. https://arxiv.org/abs/2410.13879
• KappaGCN: Bachmann et al. "Constant Curvature Graph Convolutional Networks." ICML 2020. https://proceedings.mlr.press/v119/bachmann20a.html
• ProductSpacePerceptron and ProductSpaceSVM: Tabaghi et al. "Linear Classifiers in Product Space Forms." ArXiv 2021. https://arxiv.org/abs/2102.10204
• RiemannianFuzzyKMeans and RiemannianAdan: Yuan et al. "Riemannian Fuzzy K-Means." OpenReview 2025. https://openreview.net/forum?id=9VmOgMN4Ie
• Delta-hyperbolicity computation: Based on Gromov's δ-hyperbolicity metric for tree-likeness of metric spaces. Gromov, M. "Hyperbolic Groups." Essays in Group Theory, 1987. https://link.springer.com/chapter/10.1007/978-1-4613-9586-7_3
• Sectional curvature estimation: Gu et al. "Learning Mixed-Curvature Representations in Product Spaces." ICLR 2019. https://openreview.net/forum?id=HJxeWnCcF7
• Greedy signature selection: Tabaghi et al. "Linear Classifiers in Product Space Forms." ArXiv 2021. https://arxiv.org/abs/2102.10204