index.rmd

--- 
title: "Topological Deep Learning: Going Beyond Graph Data"
author: "Mustafa Hajij, Theodore Papamarkou, Ghada Zamzmi, Karthikeyan Natesan Ramamurthy, Tolga Birdal, Michael T. Schaub"
date: "`r Sys.Date()`"
documentclass: krantz
bibliography: [bib/main.bib, bib/related-work.bib, bib/appendix.bib]
biblio-style: apalike
fontsize: 11pt
link-citations: yes
colorlinks: yes
lot: yes
lof: yes
site: bookdown::bookdown_site
description: "A book on topological deep learning."
github-repo: pyt-team/tdlbook
graphics: yes
#cover-image: images/cover.jpg
---

```{r setup, include=FALSE}
options(
  htmltools.dir.version = FALSE, formatR.indent = 2, width = 55, digits = 4
)

# install the packages needed by this book; you fill out c(), e.g. c('ggplot2', 'dplyr')
lapply(c('xfun'), function(pkg) {
  if (system.file(package = pkg) == '') install.packages(pkg)
})
```

# Contributors {-}

Along with the co-authors, the following contributors have made various contributions to the writing up of this book: *Nina Miolane, Aldo Guzmán-Sáenz, Tamal K. Dey, Soham Mukherjee, Shreyas N. Samaga, Neal Livesay, Robin Walters, Paul Rosen*.

# Preface {-}

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains.

Specifically, we first introduce *combinatorial complexes*, a novel type of topological domain. Combinatorial complexes can be seen as generalizations of graphs that maintain certain desirable properties. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations. In addition, combinatorial complexes permit the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. Thus, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces.

Second, building upon combinatorial complexes and their rich combinatorial and algebraic structure, we develop a general class of message-passing *combinatorial complex neural networks (CCNNs)*, focusing primarily on attention-based CCNNs. We characterize permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs in detail.

Third, we evaluate the performance of CCNNs on tasks related to mesh shape analysis and graph learning. Our experiments demonstrate that CCNNs have competitive performance as compared to state-of-the-art deep learning models specifically tailored to the same tasks. Our findings demonstrate the advantages of incorporating higher-order relations into deep learning models in different applications.

## Compilation {-}

This book has been compiled using the *knitr* package [@xie2015] and the *bookdown* package [@xie2024]. The source code for the book is available on [the tdlbook GitHub repository](https://github.com/pyt-team/tdlbook).

## Acknowledgments {-}

M. H. acknowledges support from the National Science Foundation, award DMS-2134231.
G. Z. is currently affiliated with National Institutes of Health (NIH), but the core of this research was done while being associated with the University of South Florida (USF). This article reports contributions of the authors and does not represent the views of NIH, or the United States Government.
N. M. acknowledges support from the National Science Foundation, Award DMS-2134241.
T. B. acknowledges support from the Engineering and Physical Sciences Research Council [grant EP/X011364/1].
T. K. D. acknowledges support from the National Science Foundation, Award CCF 2049010.
N. L. acknowledges support from the Roux Institute and the Harold Alfond Foundation.
R. W. acknowledges support from the National Science Foundation, Award DMS-2134178.
P. R. acknowledges support from the National Science Foundation, Award IIS-2316496.
M. T. S. acknowledges funding by the Ministry of Culture and Science (MKW) of the German State of North Rhine-Westphalia (NRW Rückkehrprogramm) and the European Union (ERC, HIGH-HOPeS, 101039827). Views and opinions expressed are however those of M. T. S. only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency; neither the European Union nor the granting authority can be held responsible for them.

The authors would like to thank Mathilde Papillon and Sophia Sanborn for helping improve Figure \@ref(fig:hon) and for the insightful discussions on the development of tensor diagrams.

```{block2, type='flushright', html.tag='p'}
The authors
```