TopoEmbedX: A General Framework for Representation Learning on Topological Domains

Florian Frantzen; Ibrahem AlJabea; Ines Henriques-Cadby; Theodore Papamarkou; Mustafa Hajij; Michael T. Schaub

cs.LGSep 29, 2026

TopoEmbedX: A General Framework for Representation Learning on Topological Domains

Authors: Florian Frantzen, Ibrahem AlJabea, Ines Henriques-Cadby, Theodore Papamarkou, Mustafa Hajij, Michael T. Schaub

Organizations: RWTH Aachen University · Louisiana State University · The University of Manchester · National Technical University of Athens ·

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.

Figures & tables

Panel of Figure 1 — Figure 1: Example of a topological representation learning algorithm that embeds a topological domain into a three-dimensional vector space. (a) Topological domain $\mathcal{X}$ . (b) Augmented Hasse graph $\mathcal{H}_{\mathcal{X}}(\mathsf{N})$ with cover edges, same-rank edges induced by $\mathcal{N}_{\mathrm{adj}}$ , and an example of a cross-layer edge between $0$ -cells and $2$ -cells. (c) Embedding space $\mathbb{R}^{3}$ .

Panel of Figure 2 — Figure 2: Edge classification results across different datasets (left) and edge regression results on semantic-scholar-coauth-sample (right). Missing values indicate algorithm-dataset combinations that reached the one-hour timeout described in the text.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Vertices	Maximal Faces	Classes	Imbalance Degree
algebra-questions	$423$	$1268$	$32$	$20.28$
cooking	$6714$	$39\,774$	$20$	$13.31$
geometry-questions	$580$	$1193$	$25$	$15.28$
madison-restaurant-reviews	$565$	$601$	$9$	$4.31$
MAG-10	$80\,198$	$51\,889$	$10$	$5.22$
music-blues-reviews	$1106$	$694$	$7$	$3.56$

Appendix

Table 1: Structural and label statistics for the datasets used in the edge-level experiments.

Algorithm	Hyperparameters
Cell2Vec	dimensions=128 , walk_number=10 , walk_length=80 , p=1.0 , q=1.0 , window_size=5
DeepCell	dimensions=128 , walk_number=10 , walk_length=80 , window_size=5
CellDiff2Vec	dimensions=128 , diffusion_number=10 , diffusion_cover=80 , window_size=5
HOLE	dimensions=3 , maximum_number_of_iterations=100
HOGLEE	dimensions=128
ComplexHeat	sample_number=200 , step_size=0.1 , heat_coefficient=1.0 , approximation=100 , mechanism=approximate

Appendix

Table 2: Embedding hyperparameters used in the edge-level experiments.

Algorithm	algebra-questions	cooking	geometry-questions	madison-restaurant-reviews	MAG-10	music-blues-reviews	vegas-bars-reviews
baseline_majority	$0.3294$	$0.1196$	$0.7123$	$0.6877$	$0.1838$	$0.4812$	$0.7921$
cell2vec	$0.5753$	$0.0461$	$0.7543$	$0.9193$	$0.3329$	$0.9784$	$0.9844$
celldiff2vec		$0.1866$	$0.8277$	$0.9528$			$0.9859$
deepcell	$0.8320$	$0.1972$	$0.8607$	$0.9654$	$0.2316$	$0.9921$	$0.9919$
hoglee	$0.6367$	$0.0045$	$0.7972$	$0.8136$	$0.0022$	$0.4905$	$0.7967$
hole	$0.5322$	$0.0000$	$0.7377$	$0.8996$	$0.0005$	$0.9692$	$0.9335$

Appendix

Table 3: Edge classification accuracy values. Missing values indicate algorithm-dataset combinations that reached the one-hour timeout described in Section 6 .

Algorithm	semantic-scholar-coauth-sample
baseline_mean	$89.1109$
cell2vec	$24.0678$
celldiff2vec	$49.2126$
ComplexHeat	$48.1884$
ComplexNetMF	$25.9308$
ComplexRandNE	$64.8419$

Appendix

Table 4: Edge regression mean squared error values.

Topological Representation Learning

Topological representation learning uses connectivity, holes, and higher-order structure to build or analyze machine-learning features.

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DiRe-RAPIDS: Topology-faithful dimensionality reduction at scale

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CSTEM, University of Austin, Austin, TX, USA · Department of Mathematics, Temple University, Philadelphia, PA, USA

Data Visualization Manifold Learning Dimensionality Reduction Topological Representation Learning Topological Mapping Topological Data Analysis

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Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We show that standard local metrics reward this noise memorisation: top-performing embeddings invent cycles and disconnected islands absent from the data. We introduce a topology-faithfulness benchmark based on noisy manifolds with known homology, tune DiRe against it, and find Pareto-optimal configurations that match or beat GPU-accelerated UMAP on classification while recovering exact first Betti numbers on stress tests. On 723K arXiv paper embeddings, DiRe preserves 3-4 times more topological structure than UMAP at comparable wall-clock.

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