Topological deep learning
Topological deep learning exists because molecules, meshes, and point clouds refuse to sit on a grid. Convolutional neural networks were designed for images. Recurrent networks were designed for sequences. Both depend on a shared assumption: data lives in a regular, Euclidean space. Scientific data often does not. Point clouds, meshes, scalar fields, and topological spaces like simplicial complexes and CW complexes are common in scientific computing. Standard deep learning models were not built for them. TDL draws on algebraic, differential, and geometric topology to build neural networks that can handle this kind of complexity. Its tools can capture interactions among multiple entities at once, not just pairwise binary edges. That ambition grew out of a specific problem in scientific computing. Its first expressions appeared not in topology but in the signal processing of graphs.
Geometric deep learning emerged first as a signal-processing framework for graphs. Nodes and edges define a graph, and that structure let researchers adapt convolutional operations to non-image data. Graphs have a hard limit, though: every relationship they encode is binary. An edge connects two nodes. That constraint made graphs poor tools for modeling any situation where three or more entities interact simultaneously. Follow-up work extended graph concepts to simplicial complexes and CW complexes, seeking to capture those higher-order interactions. More recent contributions proposed a unified perspective built around message-passing on general combinatorial complexes.
Topological data analysis developed a different account of structure, running in parallel with this work. TDA asked about the shape of data rather than its connectivity. Its framework was multi-scale by design, sensitive to both local and global information. Early TDA tools worked on smaller datasets and fed summaries into methods like support vector machines and random forests. Subsequent work developed coordinate systems for topological descriptors and more efficient dissimilarity measures. Those tools would later enter deep learning directly.
At the center of topological deep learning lies a question about the domain upon which data is supported. For images, that domain is a grid. For language, a sequence. TDL expands the possibilities to include fundamentally different kinds of spaces, each with its own structure and tradeoffs.
Simplicial complexes are the simplest of the higher-order domains. They extend ordinary graphs by admitting hierarchical structures. Any relation in a simplicial complex must contain all smaller subsets of that relation as part of the complex. Hodge theory can be naturally defined on simplicial complexes.
Cell complexes generalize simplicial complexes by relaxing that subset constraint. Each cell is homeomorphic to an open ball, and boundary cells of each cell must themselves be cells in the complex. The structure is represented combinatorially through incidence matrices.
Hypergraphs allow arbitrary set-type relations. A hyperedge can connect any number of entities without constraint from other relations. They do not explicitly encode the dimension of cells, providing flexibility but at the cost of some structural information.
Combinatorial complexes generalize all three. They support both hierarchical structures and set-type relations, combining the strengths of simplicial complexes, cell complexes, and hypergraphs. A rank function assigns a non-negative integer to each relation in the complex, preserving set inclusion.
Three learning tasks define the practical scope of TDL. Cell classification assigns predicted labels to individual cells, such as faces or edges in a triangular mesh. Complex classification assigns a class to an entire input mesh. Cell prediction infers properties of cell-cell interactions, or asks whether a particular cell exists in the complex. In practice, each task requires a specialized model built for the specific topological domain in question. Those models are called topological neural networks.
Topological neural networks, or TNNs, are the architectural backbone of TDL in practice. They process data on topological domains: graphs, simplicial complexes, cell complexes, and their variants. Unlike conventional neural networks designed for grids, TNNs treat the domain's topology as a core design constraint. The goal is to capture both local relationships within a cell's immediate neighborhood and global dependencies that span the broader structure.
Most TNNs use a framework called higher-order message passing. At each layer, cells exchange messages with their neighbors. What counts as a neighbor depends on the neighborhood functions defined on the domain. A message between two cells depends on the data attached to each cell and the cells' own characteristics. In cell complexes, the orientation of a cell can influence the messages it sends.
Two aggregation stages follow. First, an intra-neighborhood aggregation function combines messages within the same neighborhood. Then an inter-neighborhood aggregation function handles messages from different neighborhoods. This step allows cells that are not directly connected to exchange information if they share neighborhood relationships. A final update function combines the aggregated messages with a cell's current state to produce its representation at the next layer.
Not all TNNs follow message passing. Maggs et al. proposed an approach drawing on geometric information from embedded simplicial complexes. These are simplicial complexes with high-dimensional features attached to their vertices. That design prioritizes interpretability and geometric consistency. A separate line of work proposed a contrastive loss-based method for learning simplicial representations, offering another path that bypasses message exchange. Alongside these network-first approaches, a parallel effort was underway to integrate topological descriptors directly into neural network layers.
Hofer et al. introduced a design that opened a different route into topological deep learning. Their work added a layer to a standard deep neural network that could accept persistence diagrams and persistence barcodes as input. The key innovation was making the projection functions end-to-end trainable. This allowed topological features to be used directly in solving tasks like shape classification.
Representation learning was an early recipient of the theoretical follow-up work. Researchers examined the properties of these descriptors and connected them to broader machine learning frameworks. Other topological layers followed: some built around extended persistent homology descriptors, others around persistence landscapes or coordinate functions. Each provided a different way to pass structural information about a dataset into a neural network.
Persistent homology also proved useful in graph learning. New algorithms appeared for learning task-specific filtration functions, applicable to graph classification and node classification.
Four application areas are already establishing TDL's footprint beyond the research lab. Data compression is one. Graph neural networks are another: TDL offers tools to improve both their expressivity and their predictive performance. Action recognition and trajectory prediction round out the current application map. These four areas span a range of domains, suggesting that TDL is finding traction wherever data has complex relational structure.
TDL also encompasses methods from computational and algebraic topology that permit studying properties of neural networks themselves, including their generalization behavior. That reflexive capacity, using topology to understand the very systems doing the learning, is one of the field's less heralded possibilities.
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Common questions
What is topological deep learning?
Topological deep learning is a research field that extends deep learning to handle non-Euclidean data structures such as point clouds, meshes, simplicial complexes, and hypergraphs. It draws on algebraic, differential, and geometric topology to capture higher-order relationships and global properties. Standard models like convolutional neural networks are limited to regular grids and sequences, which most scientific data does not resemble.
What are topological neural networks used for?
Topological neural networks are designed for three broad task types in TDL: cell classification, complex classification, and cell prediction. Cell classification assigns predicted labels to individual cells such as faces or edges in a mesh. Complex classification assigns a class to an entire input mesh, and cell prediction infers properties of cell-cell interactions or predicts whether a given cell exists.
Who pioneered persistence diagrams as layers in topological deep learning networks?
Hofer et al. pioneered the integration of topological descriptors into deep neural networks. Their work introduced a layer that accepted persistence diagrams and persistence barcodes as input, using end-to-end-trainable projection functions. This approach enabled topological features to be applied to tasks like shape classification.
How does message passing work in topological neural networks?
In topological neural networks, higher-order message passing exchanges messages among cells using neighborhood functions defined on the topological domain. Messages depend on the data and characteristics of each cell, including orientation in cell complexes. Messages are aggregated first within each neighborhood, then across neighborhoods, and a final update function produces each cell's new state.
What is a simplicial complex in topological deep learning?
A simplicial complex is a higher-order domain used in topological deep learning that extends graphs by admitting hierarchical structures. Any relation in a simplicial complex must contain all smaller subsets of that relation as part of the complex. Hodge theory can be naturally defined on simplicial complexes.
What are combinatorial complexes in topological deep learning?
Combinatorial complexes are a higher-order domain in topological deep learning that generalizes simplicial complexes, cell complexes, and hypergraphs. They support both hierarchical structures and set-type relations, combining the strengths of all three. A rank function assigns a non-negative integer to each relation in the complex, preserving set inclusion.
All sources
38 references cited across the entry
- 1Topological deep learning: Going beyond graph dataM. Hajij et al. — 2022
- 2Architectures of topological deep learning: A survey on topological neural networksM. Papillon et al. — 2023
- 3Simplicial neural networksS. Ebli et al. — 2020
- 4Generalized simplicial attention neural networksC. Battiloro et al. — 2023
- 5Convolutional learning on simplicial complexesM. Yang et al. — 2023
- 6BScNets: Block Simplicial Complex Neural NetworksY. Chen et al. — 2022
- 7JournalTopological Data Analysis in smart manufacturing: State of the art and future directionsMartin Uray et al. — 2024-10-01
- 8JournalOn the Complexity of Neural Network Classifiers: A Comparison Between Shallow and Deep ArchitecturesMonica Bianchini et al. — 2014
- 9JournalTopology of Deep Neural NetworksGregory Naitzat et al. — 2020
- 10JournalIntrinsic Dimension, Persistent Homology and Generalization in Neural NetworksTolga Birdal et al. — Curran Associates, Inc. — 2021
- 11JournalPredicting the generalization gap in neural networks using topological data analysisRubén Ballester et al. — 2024
- 12JournalNeural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic TopologyBastian Rieck et al. — 2018-09-27
- 13JournalGeneralization Bounds using Data-Dependent Fractal DimensionsBenjamin Dupuis et al. — PMLR — 2023-07-03
- 14JournalGeometric Deep Learning: Going beyond Euclidean dataMichael M. Bronstein et al. — 2017
- 15JournalTopology and dataGunnar Carlsson — 2009-01-29
- 16JournalThe ring of algebraic functions on persistence bar codesAaron Adcock et al. — 2016
- 17JournalPersistence Images: A Stable Vector Representation of Persistent HomologyHenry Adams et al. — 2017
- 18JournalStatistical Topological Data Analysis using Persistence LandscapesPeter Bubenik — 2015
- 19JournalStatistical Topological Data Analysis - A Kernel PerspectiveRoland Kwitt et al. — Curran Associates, Inc. — 2015
- 20JournalSliced Wasserstein Kernel for Persistence DiagramsMathieu Carrière et al. — PMLR — 2017-07-17
- 21JournalKernel Method for Persistence Diagrams via Kernel Embedding and Weight FactorGenki Kusano et al. — 2018
- 22JournalPersistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence DiagramsTam Le et al. — Curran Associates, Inc. — 2018
- 23Cell complex neural networksM. Hajij et al. — 2020
- 24JournalSimplicial Representation Learning with Neural k-FormsKelly Maggs et al. — 2023-10-13
- 25Topo-mlp: A simplicial network without message passingK. N. Ramamurthy et al. — 2023
- 26JournalDeep Learning with Topological SignaturesChristoph Hofer et al. — Curran Associates, Inc. — 2017
- 27JournalLearning Representations of Persistence BarcodesChristoph D. Hofer et al. — 2019
- 28JournalPersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological SignaturesMathieu Carriere et al. — PMLR — 2020-06-03
- 29JournalPLLay: Efficient Topological Layer based on Persistent LandscapesKwangho Kim et al. — Curran Associates, Inc. — 2020
- 30JournalA Topology Layer for Machine LearningRickard Brüel Gabrielsson et al. — PMLR — 2020-06-03
- 31JournalGraph Filtration LearningChristoph Hofer et al. — PMLR — 2020-11-21
- 32JournalGoing beyond persistent homology using persistent homologyJohanna Immonen et al. — 2023-12-15
- 33Parametric dictionary learning for topological signal representationC. Battiloro et al. — IEEE — September 2023
- 34JournalWeisfeiler and Lehman Go Topological: Message Passing Simplicial NetworksCristian Bodnar et al. — PMLR — 2021-07-01
- 35JournalWeisfeiler and Lehman Go Cellular: CW NetworksCristian Bodnar et al. — Curran Associates, Inc. — 2021
- 36JournalTopological Graph Neural NetworksMax Horn et al. — 2021-10-06
- 37Survey of Hypergraph Neural Networks and Its Application to Action RecognitionC. Wang et al. — Springer Nature Switzerland — August 2022
- 38Principled simplicial neural networks for trajectory predictionT. M. Roddenberry et al. — PMLR — July 2021