TopoEmbedX: A General Framework for Representation Learning on Topological Domains

2026-09-29

AI Summary

This study addresses the lack of unified and efficient embedding representations for higher-order topological structures, such as simplicial complexes, by proposing the TopoEmbedX framework. Leveraging an augmented Hasse diagram expansion mechanism, this framework maps complex topological domains into Euclidean space, establishing a general representation learning paradigm that extends from graphs to multidimensional topological structures. Methodologically, it integrates classical algorithms including DeepCell and HOLE, while introducing five novel approaches based on matrix factorization and random walks, such as ComplexNetMF. Experimental results demonstrate that the generated embeddings achieve superior performance in classification and regression tasks across multidimensional data, thereby validating both the versatility and the effectiveness of the proposed framework.

Abstract

Topological structures such as simplicial complexes, hypergraphs, and cell complexes extend standard graph models by modeling higher-order relationships. These structures appear in many modern datasets and require specialized methods for generating meaningful embeddings. In this paper, we introduce TopoEmbedX, a unified framework for embedding a wide range of topological domains into Euclidean spaces. The package brings together several existing topological embedding algorithms---DeepCell, Cell2Vec, CellDiff2Vec, HOLE, and HOGLEE---and introduces five new algorithms: ComplexNetMF, ComplexRep, ComplexRandNE, ComplexWalklets, and ComplexHeat. These algorithms extend well-known graph embedding techniques to higher-order settings using the augmented Hasse graph of a topological domain. TopoEmbedX provides a clear, consistent, and easy-to-use framework for topological representation learning. Experiments show that the embeddings generated by TopoEmbedX support tasks such as classification and regression across multidimensional data.

Problem

• Topological Representation Learning
• Higher-order Relationships
• Embedding
• Simplicial Complexes
• Hypergraphs

Innovation

• Topological Representation Learning
• Higher-order Relationships
• Augmented Hasse Graph
• Topological Domains
• Graph Embedding