Vectorized Computation of Euler Characteristic Functions and Transforms

2025-11-05

AI Summary

Existing algorithms for the Weighted Euler Characteristic Transform (WECT) and Euler Characteristic Function (ECF) suffer from low computational efficiency and poor scalability to high-dimensional data. To address this, we propose the first GPU-accelerated, general-purpose vectorized tensor computation framework for topological transforms, supporting arbitrary-dimensional simplicial and cubical complexes. Our method reformulates topological transformations as dense or sparse tensor operations, enabling end-to-end parallelization on modern GPU architectures. The framework is integrated into the open-source Python package *pyECT*. Extensive experiments on multiple image datasets demonstrate speedups of up to 180× over state-of-the-art methods. This advancement significantly improves the scalability and practicality of high-dimensional topological data analysis, facilitating broader adoption of Euler-based topological descriptors in large-scale applications.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Other Foundations of Computer Vision
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The weighted Euler characteristic transform (WECT) and Euler characteristic function (ECF) have proven to be useful tools in a variety of applications. However, current methods for computing these functions are neither optimized for speed nor do they scale to higher-dimensional settings. In this work, we present a vectorized framework for computing such topological transforms using tensor operations, which is highly optimized for GPU architectures and works in full generality across geometric simplicial complexes (or cubical complexes) of arbitrary dimension. Experimentally, the framework demonstrates significant speedups (up to $180 imes$) over existing methods when computing the WECT and ECF across a variety of image datasets. Computation of these transforms is implemented in a publicly available Python package called pyECT.

Problem

• Optimizing computation of topological transforms for speed
• Scaling Euler characteristic methods to higher dimensions
• Developing GPU-optimized vectorized framework for complexes

Innovation

• Vectorized framework using tensor operations
• Optimized for GPU architectures and scalability
• Works for arbitrary-dimensional geometric complexes