Show HN: Pysimplicial, Python library for simplicial complexes in topological ML

The recent release of Pysimplicial, a Python library designed for simplicial complexes, marks a significant step in the realm of topological machine learning (ML). Developed as an experimental research toolkit, Pysimplicial aims to facilitate experiments related to simplicial triangulations and topological structures, drawing from the author’s prior research on neural networks. While still in its early development stage, this library provides a glimpse into the potential applications of topology in deep learning.

Overview of Pysimplicial

Pysimplicial is positioned as a small, experimental toolkit that emphasizes the intersection of topology and machine learning. The library is rooted in a research report entitled "Open-Closed State-sum Neural Network," which suggests a focus on the integration of topological concepts within neural network frameworks. As an experimental codebase, it is important to note that Pysimplicial is not intended for production use, indicating that it is primarily aimed at researchers and developers interested in exploring the theoretical aspects of topological deep learning.

Installation of the library is straightforward, with users able to set it up via a simple pip command:

bash
pip install git+https://github.com/kaifczxc-lab/pysimplicial.git

The library includes various functions and utilities that allow users to visualize and manipulate simplicial complexes. For instance, users can import utilities from Pysimplicial to create and visualize an octahedron, a common geometric shape in topology. The library supports operations such as relabeling vertices, applying Pachner moves (which are transformations that change the triangulation of a manifold), and calculating topological invariants like genus.

A sample workflow demonstrates how to visualize an octahedron, modify it using Pachner moves, and compute its genus. The genus is a topological property that represents the number of holes in a surface, and the library provides a method to compute this characteristic. Furthermore, users can convert the geometric figure into a feature vector suitable for multilayer perceptrons (MLPs), which are a type of artificial neural network used for various ML tasks.

Features and Functionality

Pysimplicial boasts several notable features that cater to both topological and machine learning applications. Key functionalities include:

• Pachner Moves: The library implements various Pachner moves, which are essential for manipulating triangulations in both two-dimensional and three-dimensional spaces. These moves allow users to explore different configurations of simplicial complexes.

• Triangulation Generators: Pysimplicial can generate triangulations for both 2D and 3D toroidal structures, which are fundamental in topology.

• Topological Invariants: Users can compute important topological invariants, such as genus and connected components, which provide insights into the properties of the shapes being studied.

• TQFT State-Sum: The library includes state-sum constructions based on the work of notable researchers in the field, which can be used to study two-dimensional open-closed Topological Quantum Field Theories (TQFT).

• Converters for ML Models: Pysimplicial offers converters that facilitate the integration of topological features into various machine learning models, including Graph Neural Networks (GNNs), Tensor Neural Networks (TNNs), and MLPs. This feature is particularly significant for researchers looking to leverage topological data in their ML applications.

Documentation for Pysimplicial is accessible through a dedicated showcase notebook, which provides examples of how to utilize its functions effectively. Additionally, the library encourages contributions from the community, indicating a collaborative approach to its development.

Why it matters

The introduction of Pysimplicial is a noteworthy development for creators and technologists working at the intersection of machine learning and topology. As the field of topological ML continues to evolve, tools like Pysimplicial can empower researchers to explore complex geometric structures and their implications for data analysis and modeling.

For creators, this library opens up new avenues for experimentation with topological data, allowing for innovative approaches to problem-solving in ML. The ability to manipulate and visualize simplicial complexes can lead to deeper insights into the relationships between data points, potentially enhancing model performance.

For technologists, Pysimplicial serves as a reminder of the importance of interdisciplinary collaboration. By bridging the gap between topology and machine learning, this library encourages the integration of mathematical concepts into computational frameworks, paving the way for novel applications and methodologies in AI.

In conclusion, while Pysimplicial is still in its early stages, its potential impact on the fields of topological ML and data science is significant. As researchers continue to explore the capabilities of this toolkit, it may very well contribute to the advancement of knowledge and techniques in both topology and machine learning.