Chapter 10 Related work

Topological deep learning (TDL) has recently emerged as a new research frontier that lies at the intersection of several areas, including geometric and topological machine learning, and network science. To demonstrate where TDL fits in the existing literature, we review a broad spectrum of prior works, and categorize them into graph-based models, higher-order deep learning models, graph-based pooling, attention-based models, and applied algebraic topology.

10.1 Graph-based models

Graph-based models have been widely used for modeling pairwise interactions (edges) between elements (vertices) of different systems, including social systems (e.g., social network analysis) and biological systems (e.g., protein-protein interactions), see (Knoke and Yang 2019; Jha, Saha, and Singh 2022). Based on their edge or vertex properties, graphs can be classified as unweighted graphs (unweighted edges), weighted graphs (weighted edges), signed graphs (signed edges), undirected or directed graphs (undirected or directed edges), and spatio-temporal graphs (spatio-temporal vertices), as discussed in (Goyal and Ferrara 2018; Zonghan Wu et al. 2020). Each of these graph types can be combined with neural networks to form graph neural networks and model different interactions in various systems (Goyal and Ferrara 2018; Zonghan Wu et al. 2020). For example, unweighted and undirected graph-based models have been used for omic data mapping (Amar and Shamir 2014) and mutual friendship detection in social networks (Tabassum et al. 2018); weighted graph-based models have been widely used with systems related to traffic forecasting (Halaoui 2010; Q. Zhang et al. 2018) and epidemiological modeling/forecasting (Linka et al. 2020; Manrı́quez, Guerrero-Nancuante, and Taramasco 2021); signed graph-based models are suitable for tasks such as segmentation (Bailoni et al. 2022) and clustering (Kunegis et al. 2010; Gallier 2016); spatio-temporal graph-based models can describe systems that are spatio-temporal in nature, such as human activity and different types of motion (Yan, Xiong, and Lin 2018; Bhattacharya et al. 2020; Plizzari, Cannici, and Matteucci 2021).

As graph-based approaches that utilize single-layer or monolayer graphs cannot model multiple types of relations between vertices in a network (Goyal and Ferrara 2018; Zonghan Wu et al. 2020), multilayer or multiplex networks have been proposed (Kivelä et al. 2014; W. Zhang et al. 2020; Chang et al. 2022). Similar to monolayer graphs, multiplex networks contain vertices and edges, but the edges exist in separate layers, where each layer represents a specific type of interaction or relation. Multiplex networks have been used in various applications, including multilayer modeling of the human brain (De Domenico 2017; Anand and Chung 2023) and online gaming (Chang et al. 2022). All these types of networks can only model pairwise relations between vertices, motivating the need for higher-order networks, as discussed in Section 2.2.

10.2 Higher-order deep learning models

In recent years, there has been an increasing interest in higher-order networks (Mendel 1991; Battiston et al. 2020; Bick et al. 2021) due to the ability of these networks to adequately capture higher-order interactions. Hodge-theoretic approaches, message passing schemes, and skip connections have been developed for higher-order networks in the signal processing and deep learning literature.

A Hodge-theoretic approach (Lim 2020) over simplicial complexes has been introduced by (Barbarossa and Sardellitti 2020a; Schaub et al. 2021). This effort has been extended to hypergraphs by (Barbarossa and Tsitsvero 2016; Schaub et al. 2021) and to cell complexes by (T. Mitchell Roddenberry, Schaub, and Hajij 2022; Sardellitti, Barbarossa, and Testa 2021). The work of (T. Mitchell Roddenberry and Segarra 2019) has defined an edge-based convolutional neural network by exploiting the 1-Hodge Laplacian operator for linear filtering (Barbarossa, Sardellitti, and Ceci 2018; Schaub and Segarra 2018; Barbarossa and Sardellitti 2020a, 2020b; Schaub et al. 2021).

Convolutional operators and message-passing algorithms have been developed for higher-order neural networks. For example, a convolutional operator on hypergraphs has been proposed by (Arya and Worring 2018; Feng et al. 2019; J. Jiang et al. 2019) and has been investigated further by (J. Jiang et al. 2019; Y. Gao et al. 2020; S. Bai, Zhang, and Torr 2021; J. Bai et al. 2021; L. Giusti, Battiloro, Testa, et al. 2022; H. Wu and Ng 2022; Gong, Higham, and Zygalakis 2023). A unifying framework for learning on graphs and hypergraphs has been proposed recently in (Jing Huang and Yang 2021). The authors in (Y. Gao et al. 2022) have introduced the so-called general hypergraph neural networks, which constitute a multi-modal/multi-type data correlation modeling framework. As for message passing on complexes, the work of (Hajij, Istvan, and Zamzmi 2020) has introduced a higher-order message-passing framework that encompasses those proposed by (Gilmer et al. 2017; Bunch et al. 2020; Ebli, Defferrard, and Spreemann 2020; Hayhoe et al. 2022) and has utilized various local neighborhood aggregation schemes. In (E. C. Mitchell et al. 2022), recurrent simplicial neural networks have been proposed and applied to trajectory prediction. The authors in (Calmon, Schaub, and Bianconi 2022) have addressed the challenge of processing signals supported on multiple cell dimensions concurrently, by introducing a coupling multi-signal approach on higher-order networks that utilizes the Dirac operator. Several simplicial and cellular neural networks have been introduced recently, including (Burns and Fukai 2023; Bodnar et al. 2021; T. Mitchell Roddenberry, Schaub, and Hajij 2022; Sardellitti, Barbarossa, and Testa 2021; Sardellitti and Barbarossa 2022; Battiloro et al. 2023; Yang and Isufi 2023). For more details, the reader is referred to the recent survey of (Papillon et al. 2023) on TDL.

A generalization of skip connections (Ronneberger, Fischer, and Brox 2015; He et al. 2016) to simplicial complexes has been introduced by (Hajij, Ramamurthy, et al. 2022), which allows the training of higher-order deep neural networks. The authors in (Morris et al. 2019) have proposed a higher-order graph neural network that takes into account higher-order graph structures at multiple scales. While these methods allow for multi-way hierarchical coupling, the coupling is isotropic and weight differences within a particular multi-way connection can not be learned. These limitations can be alleviated by attention-based models.

Higher-order models have achieved promising performance in several real-world applications, including link prediction (Hajij, Ramamurthy, et al. 2022; Piaggesi, Panisson, and Petri 2022; Yuzhou Chen, Gel, and Poor 2022), action recognition (C. Wang et al. 2023), visual classification (Shi et al. 2018), optimal homology generator detection (Keros, Nanda, and Subr 2022), time series (Andrea Santoro et al. 2023), dynamical systems (Majhi, Perc, and Ghosh 2022), spectral clustering (Reddy, Chepuri, and Borgnat 2023), node classification (Hajij, Ramamurthy, et al. 2022), and trajectory prediction (Benson et al. 2018; T. Mitchell Roddenberry, Glaze, and Segarra 2021).

10.3 Attention-based models

Real-world relational data is large, unstructured, sparse and noisy. As a result, graph neural networks (GNNs) may learn suboptimal data representations, and therefore may exhibit compromised performance (Zonghan Wu et al. 2020; Asif et al. 2021; Dai, Aggarwal, and Wang 2021). To address these issues, various attention mechanisms (Chaudhari et al. 2021) have been incorporated in GNNs, which allow to learn neural architectures that detect the most relevant parts of a given graph while ignoring irrelevant parts. Based on the used attention mechanism, existing graph attention approaches can be divided into weight-based attention, similarity-based attention, and attention-guided walk (J. B. Lee et al. 2019).

The majority of attention-based mechanisms, with the exception of (S. Bai, Zhang, and Torr 2021; E.-S. Kim et al. 2020; Georgiev, Brockschmidt, and Allamanis 2022; L. Giusti, Battiloro, Testa, et al. 2022; L. Giusti, Battiloro, Di Lorenzo, et al. 2022; Goh, Bodnar, and Lio 2022), are designed for graphs. For example, the attention model proposed by (Goh, Bodnar, and Lio 2022) is a generalization of the graph attention model of (Veličković et al. 2018). In (L. Giusti, Battiloro, Di Lorenzo, et al. 2022), the authors have utilized a model based on Hodge decomposition, similar to the one suggested in (T. Mitchell Roddenberry, Glaze, and Segarra 2021), to introduce an attention model for simplicial complexes. The hypergraph attention models introduced in (E.-S. Kim et al. 2020; S. Bai, Zhang, and Torr 2021) provide alternative generalizations of the graph attention model of (Veličković et al. 2018). The aforementioned attention models neither allow nor combine higher-order attention blocks of entities of different dimensions. This limits the space of neural architectures and the scope of applications of existing attention models.

10.4 Graph-based pooling

Several attempts have been made to emulate the success of image-based pooling layers in the context of graphs. Some of the early work employs popular graph clustering algorithms (Kushnir, Galun, and Brandt 2006; Dhillon, Guan, and Kulis 2007) to achieve graph-based pooling architectures (Bruna et al. 2014). Coarsening operations have been applied to graphs to attain the invariance properties needed in learning tasks (Ying et al. 2018; Mesquita, Souza, and Kaski 2020; H. Gao, Liu, and Ji 2021). The current state-of-the-art graph-based pooling approaches mostly rely on dynamically learning the pooling needed for the learning task (Grattarola et al. 2022). This includes spectral methods (Ma et al. 2019), clustering methods such as DiffPool (Ying et al. 2018) and MinCut (Bianchi, Grattarola, and Alippi 2020), top-K methods (H. Gao and Ji 2019; J. Lee, Lee, and Kang 2019; Z. Zhang et al. 2021), and hierarchical graph pooling (Jingjia Huang et al. 2019; J. Lee, Lee, and Kang 2019; Z. Zhang et al. 2019, 2021; J. Li et al. 2020; Pang, Zhao, and Li 2021). Pooling on higher-order networks remains unstudied, with the exception of a general simplicial complex pooling strategy developed by (Cinque, Battiloro, and Di Lorenzo 2022) along the lines of the proposal made by (Grattarola et al. 2022).

10.5 Applied algebraic topology

Although algebraic topology (Hatcher 2005) is a relatively old field, applications of this field have only recently started to crystallize (G. Carlsson 2009; Edelsbrunner and Harer 2010). Indeed, topological constructions have been found to be natural tools for the formulation of longstanding problems in many fields. For instance, persistent homology (Edelsbrunner and Harer 2010) has been successful at finding solutions to various complex data problems (Boyell and Ruston 1963; Kweon and Kanade 1994; Bajaj, Pascucci, and Schikore 1997; Attene, Biasotti, and Spagnuolo 2003; Carr, Snoeyink, and Panne 2004; H. Lee et al. 2011b, 2011a, 2012a, 2012b; Dabaghian et al. 2012; Nicolau, Levine, and Carlsson 2011; Lum et al. 2013; C. Giusti, Ghrist, and Bassett 2016; Curto 2017; Rosen et al. 2017). Recent years have witnessed increased interest in the role of topology in machine learning and data science (Hensel, Moor, and Rieck 2021; Dey and Wang 2022b). Topology-based machine learning models have been applied in many areas, including topological signatures of data (Biasotti et al. 2008; G. Carlsson et al. 2005; Rieck and Leitte 2015), neuroscience (H. Lee et al. 2011b, 2011a, 2012a, 2012b; Dabaghian et al. 2012; C. Giusti, Ghrist, and Bassett 2016; Curto 2017), bioscience (DeWoskin et al. 2010; Nicolau, Levine, and Carlsson 2011; Chan, Carlsson, and Rabadan 2013; Taylor et al. 2015; Topaz, Ziegelmeier, and Halverson 2015; Lo and Park 2016), the study of graphs (Horak, Maletić, and Rajković 2009; Weinan, Jianfeng, and Yuan 2013; Bampasidou and Gentimis 2014; Carstens and Horadam 2013; Petri et al. 2013a, 2013b; Rieck, Bock, and Borgwardt 2019; Hajij and Rosen 2020), time series forcasting (Zeng et al. 2021), Trojan detection (Hu et al. 2022), image segmentation (Hu et al. 2019), 3D reconstruction (Waibel et al. 2022), and time-varying setups (Edelsbrunner et al. 2004; Perea et al. 2015; Maletić, Zhao, and Rajković 2016; Rieck et al. 2020).

Topological data analysis (TDA) (G. Carlsson 2009; Edelsbrunner and Harer 2010; Ghrist 2014; Love et al. 2023b; Dey and Wang 2022b) has emerged as a scientific area that harnesses topological tools to analyze data and develop machine learning algorithms. TDA has found many applications in machine learning, including enhancing existing machine learning models (BenTaieb and Hamarneh 2016; Hofer et al. 2017; Clough et al. 2019; Gabrielsson et al. 2020; F. Wang et al. 2020; Leventhal et al. 2023), improving the explainability of deep learning models (G. Carlsson and Gabrielsson 2020; Elhamdadi, Canavan, and Rosen 2021; Love et al. 2023a), dimensionality reduction (Moor et al. 2020), filtration learning (Hofer et al. 2020), and topological layers constructions (K. Kim et al. 2020). A notable research trend has been the vectorization of persistence diagrams. Vector representations of persistence diagrams are constructed in order to be utilized in downstream machine learning tasks. These methods include Betti curves (Umeda 2017), persistence landscapes (Bubenik 2015), persistence images (Adams et al. 2017), and other vectorization constructions (Y.-C. Chen et al. 2015; Kusano, Hiraoka, and Fukumizu 2016; Berry et al. 2020). A unification of these methods has been proposed recently in (Carriere et al. 2020).

Our work introduces combinatorial complexes (CCs) as a generalized higher-order network on which deep learning models can be defined and studied in a unifying manner. Hence, our work expands TDA by formalizing deep learning notions in topological terms and by realizing constructions in TDA, e.g., mapper (Singh et al. 2007), in terms of our TDL framework. The construction of CCs and of combinatorial complex neural networks (CCNNs), which are neural networks defined on CCs, is inspired by classical notions in algebraic topology (Hatcher 2005) and in topological quantum field theory (Turaev 2016), and by recent advances in TDA (Collins et al. 2004; G. Carlsson et al. 2005, 2008; E. Carlsson, Carlsson, and De Silva 2006; G. Carlsson and Mémoli 2008; G. Carlsson and Zomorodian 2009; G. Carlsson 2009) as applied to machine learning (Pun, Xia, and Lee 2018; Dey and Wang 2022b).

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