Topological deep learning

Topological deep learning (TDL)  represents a field at the intersection of topology and deep learning, offering approaches to analyze and learn from data structured in topological spaces. By leveraging the principles of topology, TDL offers an approach to understanding and processing data supported on topological spaces.

Motivation
Conventional deep learning often operates under the assumption that the data under examination reside in a linear vector space and can be effectively characterized using feature vectors. However, there is a growing recognition that this conventional perspective may be inadequate for describing various real-world datasets. For instance, molecules are more aptly represented as graphs rather than feature vectors. Similarly, three-dimensional objects, such as those encountered in computer graphics and geometry processing, are better represented as meshes. Additionally, data originating from social networks, where actors are interconnected in complex ways, defy simple vector-based descriptions. Consequently, there has been a surge in interest in integrating concepts from topology into traditional deep learning frameworks to gain more accurate structural representation of the underlying data.

An introduction to topological domains
One of the core concepts in topological deep learning is the domain upon which this data is defined and supported. In case of Euclidian data, such as images, this domain is a grid, upon which the pixel value of the image is supported. In a more general setting this domain might be a topological domain. Next, we introduce the most common topological domains that are encountered in a deep learning setting. These domains include, but not limited to, graphs, simplicial complexes, cell complexes, combinatorial complexes and hypergraphs.

Given a finite set S of abstract entities, a neighborhood function $$\mathcal{N}$$ on S is an assignment that attach to every point $$x$$ in S a subset of S or a relation. Such a function can be induced by equipping S with an auxiliary structure. Edges provide one way of defining relations among the entities of S. More specifically, edges in a graph allow one to define the notion of neighborhood using, for instance, the one hop neighborhood notion. Edges however, limited in their modeling capacity as they can only be used to model binary relations among entities of S since every edge is connected typically to two entities. In many applications, it is desirable to permit relations that incorporate more than two entities. The idea of using relations that involve more than two entities is central to topological domains. Such higher-order relations allow for a broader range of neighborhood functions to be defined on S to capture multi-way interactions among entities of S.

Next we review the main properties, advantages, and disadvantages of some commonly studied topological domains in the context of deep learning, including (abstract) simplicial complexes, regular cell complexes, hypergraphs, and combinatorial complexes.

Comparisons among topological domains
Each of the enumerated topological domains has its own characteristics, advantages, and limitations:


 * Simplicial complexes
 * Simplest form of higher-order domains.
 * Extensions of graph-based models.
 * Admit hierarchical structures, making them suitable for various applications.
 * Hodge theory can be naturally defined on simplicial complexes.
 * Require relations to be subsets of larger relations, imposing constraints on the structure.


 * Cell Complexes
 * Generalize simplicial complexes.
 * Provide more flexibility in defining higher-order relations.
 * Each cell in a cell complex is homeomorphic to an open ball, attached together via attaching maps.
 * Boundary cells of each cell in a cell complex are also cells in the complex.
 * Represented combinatorially via incidence matrices.


 * Hypergraphs
 * Allow arbitrary set-type relations among entities.
 * Relations are not imposed by other relations, providing more flexibility.
 * Do not explicitly encode the dimension of cells or relations.
 * Useful when relations in the data do not adhere to constraints imposed by other models like simplicial and cell complexes.


 * Combinatorial Complexes :
 * Generalize and bridge the gaps between simplicial complexes, cell complexes, and hypergraphs.
 * Allow for hierarchical structures and set-type relations.
 * Combine features of other complexes while providing more flexibility in modeling relations.
 * Can be represented combinatorially, similar to cell complexes.

Hierarchical structure and set-type relations
The properties of simplicial complexes, cell complexes, and hypergraphs give rise to two main features of relations on higher-order domains, namely hierarchies of relations and set-type relations.

Rank function
A rank function on a higher-order domain X is an order-preserving function rk: X → Z, where rk(x) attaches a non-negative integer value to each relation x in X, preserving set inclusion in X. Cell and simplicial complexes are common examples of higher-order domains equipped with rank functions and therefore with hierarchies of relations.

Set-type relations
Relations in a higher-order domain are called set-type relations if the existence of a relation is not implied by another relation in the domain. Hypergraphs constitute examples of higher-order domains equipped with set-type relations. Given the modeling limitations of simplicial complexes, cell complexes, and hypergraphs, we develop the combinatorial complex, a higher-order domain that features both hierarchies of relations and set-type relations.

Learning on topological spaces


The learning tasks in TDL can be broadly classified into three categories:


 * Cell classification: Predict targets for each cell in a complex. Examples include triangular mesh segmentation, where the task is to predict the class of each face or edge in a given mesh.
 * Complex classification: Predict targets for an entire complex. For example, predict the class of each input mesh.
 * Cell prediction: Predict properties of cell-cell interactions in a complex, and in some cases, predict whether a cell exists in the complex. An example is the prediction of linkages among entities in hyperedges of a hypergraph.

In practice, to perform the aforementioned tasks, deep learning models designed for specific topological spaces must be constructed and implemented. These models, known as topological neural networks, are tailored to operate effectively within these spaces.

Topological neural networks
Central to TDL are topological neural networks (TNNs), specialized architectures designed to operate on data structured in topological domains. Unlike traditional neural networks tailored for grid-like structures, TNNs are adept at handling more intricate data representations, such as graphs, simplicial complexes, and cell complexes. By harnessing the inherent topology of the data, TNNs can capture both local and global relationships, enabling nuanced analysis and interpretation.

Message passing topological neural networks
In a general topological domain, higher-order message passing involves exchanging messages among entities and cells using a set of neighborhood functions.

Definition: Higher-Order Message Passing on a General Topological Domain

Let $$\mathcal{X}$$ be a topological domain. We define a set of neighborhood functions $$\mathcal{N}=\{ \mathcal{N}_1,\ldots,\mathcal{N}_n\}$$ on $$\mathcal{X}$$. Consider a cell $$x$$ and let $$y\in \mathcal{N}_k(x)$$ for some $$\mathcal{N}_k \in \mathcal{N}$$. A message $$m_{x,y}$$ between cells $$x$$ and $$y$$ is a computation dependent on these two cells or the data supported on them. Denote $$\mathcal{N}(x)$$ as the multi-set $$\{\!\!\{ \mathcal{N}_1(x), \ldots , \mathcal{N}_n (x) \}\!\!\}$$, and let $$\mathbf{h}_x^{(l)}$$ represent some data supported on cell $$x$$ at layer $$l$$. Higher-order message passing on $$\mathcal{X}$$, induced by $$\mathcal{N}$$, is defined by the following four update rules:


 * 1) $$m_{x,y} = \alpha_{\mathcal{N}_k}(\mathbf{h}_x^{(l)},\mathbf{h}_y^{(l)})$$
 * 2) $$m_{x}^k =  \bigoplus_{y \in \mathcal{N}_k(x)}  m_{x,y}$$, where $$\bigoplus$$ is the intra-neighborhood aggregation function.
 * 3) $$m_{x} =  \bigotimes_{ \mathcal{N}_k \in \mathcal{N} } m_x^k$$, where $$\bigotimes$$ is the inter-neighborhood aggregation function.
 * 4) $$\mathbf{h}_x^{(l+1)} = \beta (\mathbf{h}_x^{(l)}, m_x)$$, where $$\alpha_{\mathcal{N}_k},\beta$$ are differentiable functions.

Some remarks on Definition above are as follows.

First, Equation 1 describes how messages are computed between cells $$x$$ and $$y$$. The message $$m_{x,y}$$ is influenced by both the data $$\mathbf{h}_x^{(l)}$$ and $$\mathbf{h}_y^{(l)}$$ associated with cells $$x$$ and $$y$$, respectively. Additionally, it incorporates characteristics specific to the cells themselves, such as orientation in the case of cell complexes. This allows for a richer representation of spatial relationships compared to traditional graph-based message passing frameworks.

Second, Equation 2 defines how messages from neighboring cells are aggregated within each neighborhood. The function $$\bigoplus$$ aggregates these messages, allowing information to be exchanged effectively between adjacent cells within the same neighborhood.

Third, Equation 3 outlines the process of combining messages from different neighborhoods. The function $$\bigotimes$$ aggregates messages across various neighborhoods, facilitating communication between cells that may not be directly connected but share common neighborhood relationships.

Fourth, Equation 4 specifies how the aggregated messages influence the state of a cell in the next layer. Here, the function $$\beta$$ updates the state of cell $$x$$ based on its current state $$\mathbf{h}_x^{(l)}$$ and the aggregated message $$m_x$$ obtained from neighboring cells.

Non-message passing topological neural networks
While the majority of TNNs follow the message passing paradigm. A few models have been suggested that do not fall this model. For instance, in leverages geometric information from simplicial complexes embedded in multidimensional spaces using node coordinates. This offers interpretability and geometric consistency without relying on message passing. Furthermore, in a contrastive loss-based method was suggested to learn the simplicial representation.

Applications
TDL is rapidly finding new applications in across various fields including data compression, enhancing the expressive capacity of graph neural networks, action recognition, and trajectory prediction.