Topological Deep Learning Is Not Topological Data Analysis

Two Things Called “Topology”

Read enough papers with “topological” in the title and you will notice they do not describe the same activity.

One kind of paper takes a point cloud of protein conformations, builds a Vietoris–Rips filtration, computes persistent homology, turns the resulting diagram into a persistence image, concatenates that vector with some chemical descriptors, and trains a gradient-boosted tree. Topology appears once, at the start, and then never again. The model is a tree; it has no idea what a homology class is.

Another kind of paper takes a molecule, treats its six-membered rings as two-dimensional cells rather than as six separate edges, and defines a neural network whose layers pass messages between vertices, edges, and rings according to which contains which. There is no persistence diagram anywhere. There is no filtration. But the domain the network operates on is a cell complex, and every layer respects that structure.

Both are legitimate. They are also almost entirely different techniques, with different failure modes, different computational costs, and different things they can and cannot represent. Book IV is titled Topological Deep Learning while most of its chapters describe the first programme. This post draws the line.

Pipeline One: Compute, Then Learn

The classical TDA pipeline is a sequence of four stages, and the crucial property is that only the last one has parameters.

You choose a filtration (Rips, alpha, sublevel sets of a function, cubical for images). You run persistent homology to get a multiset of birth–death pairs. You apply a vectorisation map \(\Phi\) — a persistence image, a persistence landscape, a Betti curve, a vector of persistence entropies — to land in a fixed-dimensional Euclidean space. Then you train whatever you like: an SVM, a random forest, an MLP.

The properties of this pipeline follow directly from its shape:

• The descriptor is fixed before training. \(\Phi(\mathrm{dgm}(X))\) does not depend on \(\theta\), on the labels, or on the loss. If the topology you computed is not the topology the task needs, no amount of training fixes it.
• The learning happens in \(\mathbb{R}^d\). Once you have the vector, you are doing ordinary supervised learning. Nothing downstream is topological.
• You inherit stability for free. The bottleneck stability theorem bounds how far the diagram can move when the input moves, and good vectorisations are Lipschitz on top of that. This is a real guarantee, and it is the main reason the pipeline is worth using.
• It works when \(n\) is small. A persistence image is a hand-designed feature map with no parameters. On datasets of a few hundred samples, that is an advantage, not a limitation.

The honest summary: this is feature engineering, done with unusually good mathematics.

Pipeline Two: Learning on a Topological Domain

Now change what the input is.

Suppose the data point is not a point cloud but a simplicial complex \(K\): a set of vertices, edges, triangles, tetrahedra, closed under taking faces. Attach a feature vector to every simplex, not just every vertex. Define a layer that updates the feature of a simplex \(\sigma\) from the features of the simplices it touches — its faces, its cofaces, and the simplices it is adjacent to through a shared face or a shared coface. Stack those layers. Read out.

Schematically, for a \(k\)-simplex \(\sigma\) with features \(h_\sigma\):

where \(\mathcal{B}(\sigma)\) are the boundary simplices (faces of \(\sigma\)), \(\mathcal{C}(\sigma)\) the coboundary simplices (\(\sigma\) is a face of them), \(\mathcal{N}_{\uparrow}(\sigma)\) the upper-adjacent simplices (those sharing a coface with \(\sigma\)), and \(\bigoplus\) a permutation-invariant aggregator. Set \(k = 0\), delete every term except upper adjacency, and you have recovered an ordinary graph neural network. That is not a coincidence; it is the point.

Nothing here computes homology. No filtration is chosen. The topology is in the incidence structure that the messages travel along.

A Classification Table

What a Graph Cannot Say

Here is the concrete reason the second programme exists.

A graph edge is a pair. That is not a convention; it is what an edge is. So whenever the real relation in your data involves three or more entities at once — three authors on one paper, six atoms in one aromatic ring, four nodes in one communication group, a triangular face of a mesh — the graph representation must encode it indirectly. The standard trick is the clique expansion: replace each group by all the pairwise edges among its members.

The clique expansion loses information, and it is easy to see exactly how.

The minimal example above is deliberately blunt: one triple versus three pairs. A sceptic will say the two hypergraphs differ in the size of their hyperedges, so of course a graph loses that. So here is a second pair, in which every hyperedge has size exactly three.

Expand \(H_A\): the triple \(\{1,2,3\}\) contributes edges 12, 13, 23; \(\{1,2,4\}\) contributes 12, 14, 24; \(\{1,3,4\}\) contributes 13, 14, 34. The union is \(\{12, 13, 14, 23, 24, 34\}\) — all six edges, so the clique expansion is \(K_4\). Expand \(H_B\): it contains every triple of \(H_A\), so its expansion contains \(K_4\), and \(K_4\) is all there is. Both clique-expand to \(K_4\).

They are not isomorphic hypergraphs: \(H_A\) has 3 hyperedges and \(H_B\) has 4, and their vertex degree sequences are \((2,2,2,3)\) and \((3,3,3,3)\). So this is not a relabelling artefact. Two genuinely different 3-uniform hypergraphs, one graph.

The lift keeps what the expansion discards

Take the downward closure of each hypergraph — add every subset of every hyperedge — and you get a simplicial complex. Count the simplices:

• \(H_A\) closed: 4 vertices, 6 edges, 3 triangles. 13 simplices. Euler characteristic \(\chi = 4 - 6 + 3 = 1\).
• \(H_B\) closed: 4 vertices, 6 edges, 4 triangles. 14 simplices. Euler characteristic \(\chi = 4 - 6 + 4 = 2\).
• The clique complex of \(K_4\) (every clique becomes a simplex): 4 vertices, 6 edges, 4 triangles, 1 tetrahedron. 15 simplices. \(\chi = 4 - 6 + 4 - 1 = 1\).

Three different complexes, one shared 1-skeleton. And the difference is not cosmetic. The closure of \(H_B\) is the hollow tetrahedron \(\partial \Delta^3\), which is a 2-sphere: \(\beta_2 = 1\). The closure of \(H_A\) is the cone with apex 1 over the cycle \(2 \to 3 \to 4 \to 2\), which is contractible: \(\beta_2 = 0\). One encloses a void, the other does not, and their graphs are indistinguishable.

The Line Genuinely Blurs in Two Places

Being honest about where the distinction breaks down is more useful than defending it too hard.

A graph is already a topological object. A graph is a 1-dimensional simplicial complex. Its \(\beta_0\) counts connected components and its \(\beta_1\) counts independent cycles; these are homology groups, computed on a chain complex, in the ordinary sense. So a GNN is already doing message passing on a topological domain — one that happens to be truncated at dimension 1. “Graphs versus topology” is a difference of degree, not of kind, and the honest framing of TDL is the same idea, without the truncation. This is also why the expressivity story is continuous rather than a jump: simplicial and cellular Weisfeiler–Leman refinements are defined by the same colour-refinement recipe as 1-WL, run over cells of every dimension instead of vertices only.

Differentiable persistence deliberately erases the boundary. If the filtration values are a differentiable function of learnable parameters, and the persistence diagram is a piecewise-linear function of those values, then \(\Phi(\mathrm{dgm})\) becomes a differentiable layer and the whole “preprocessing” stage joins the computation graph. At that point topology is no longer fixed before training — the model learns which filtration to build. Topological autoencoders and topological regularisation losses live here. They are TDA machinery inside a TDL training loop, and calling them one or the other is a matter of emphasis.

There is no contradiction in using both at once. Nothing stops you from running a simplicial network on a lifted complex and appending a persistence-image feature to the readout. They answer different questions: persistence tells you about the shape of the data at many scales; higher-order message passing tells you how to compute on a fixed shape.

What Higher-Order Structure Does Not Buy You

Four things worth saying plainly, because the enthusiastic version of this post would skip them.

The lift can be enormous. If you lift a graph by filling in all its cliques, the count is combinatorial. On \(K_{20}\) there are \(\binom{20}{2} = 190\) edges but \(\binom{20}{3} = 1140\) triangles and \(\binom{20}{4} = 4845\) tetrahedra. Dense subgraphs are exactly where clique lifting explodes, and social and citation graphs are full of them. Ring lifting and other sparse lifts exist precisely because of this.

The lift is a modelling assumption, not data. If your dataset is a plain graph, the higher-order structure did not come with it — you invented it. A clique lift and a ring lift on the same molecule produce different models with different inductive biases, and the choice is on you. This is the exact counterpart of choosing a filtration in the TDA pipeline; the arbitrariness did not go away, it moved.

More expressive does not mean better generalisation. The theory says a strictly larger set of inputs can be told apart. It says nothing about test error. An architecture that can distinguish more graphs can also overfit distinctions that are noise.

Many reported gains are domain-specific. The clean win cases for higher-order message passing are the ones where the group structure is real and load-bearing — rings in molecules, faces in meshes, co-authorship in bibliographic data, simplicial signals like flows on edges. On a graph whose relations genuinely are pairwise, lifting adds cost and buys nothing.

How to Read the Rest of Book IV

Most chapters in this book — filtrations, persistence diagrams, stability, the twist algorithm, Mapper, persistence images, the application chapters — are Pipeline One. They are TDA, and the deep learning in them is a downstream consumer.

A smaller set — differentiable persistence, learning filtrations, topological autoencoders, topological regularisation — sits on the boundary, using TDA objects inside a gradient-based loop.

The chapters that would be TDL proper are the ones about computing on a complex: simplicial and cellular networks, hypergraph networks, sheaf neural networks, higher-order attention. If you are reading this book to learn how to build a model that consumes a simplicial complex, this post is the map that tells you which chapters will actually help.

References

• Xu, K., Hu, W., Leskovec, J., & Jegelka, S. (2019). How Powerful are Graph Neural Networks? ICLR 2019 (establishes 1-WL as the expressivity ceiling for message-passing GNNs — the baseline that higher-order methods aim to exceed).
• Bronstein, M. M., Bruna, J., Cohen, T., & Veličković, P. (2021). Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (the symmetry-first framing in which “choose the domain, then the equivariant layer” is the design principle TDL follows).
• Bodnar, C., Frasca, F., Wang, Y. G., Otter, N., Montúfar, G., Liò, P., & Bronstein, M. (2021). Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks. ICML 2021 (introduces simplicial WL and MPSN; the strict improvement over 1-WL).
• Bodnar, C., Frasca, F., Otter, N., Wang, Y. G., Liò, P., Montúfar, G., & Bronstein, M. (2021). Weisfeiler and Lehman Go Cellular: CW Networks. NeurIPS 2021 (extends the construction to regular cell complexes, where a ring is a single 2-cell).
• Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz, A., Ramamurthy, K. N., Birdal, T., Dey, T. K., Mukherjee, S., Samaga, S. N., Livesay, N., Walters, R., Rosen, P., & Schaub, M. T. (2022). Topological Deep Learning: Going Beyond Graph Data. arXiv:2206.00606 (combinatorial complexes as a common generalisation of simplicial complexes, cell complexes, and hypergraphs).
• Carlsson, G. (2009). Topology and Data. Bulletin of the American Mathematical Society, 46(2) (the survey that set out the persistent-homology-as-feature-extraction programme).
• Bubenik, P. (2015). Statistical Topological Data Analysis using Persistence Landscapes. Journal of Machine Learning Research (a Banach-space vectorisation of persistence diagrams).
• Adams, H., et al. (2017). Persistence Images: A Stable Vector Representation of Persistent Homology. Journal of Machine Learning Research, 18(8), 1–35 (the most widely used diagram vectorisation).
• Zhou, D., Huang, J., & Schölkopf, B. (2006). Learning with Hypergraphs: Clustering, Classification, and Embedding. NIPS 2006 (the source of the normalised clique-expansion Laplacian that most hypergraph methods still reduce to).
• Ebli, S., Defferrard, M., & Spreemann, G. (2020). Simplicial Neural Networks. arXiv:2010.03633 (convolution on simplicial complexes via the Hodge Laplacian).
• Feng, Y., You, H., Zhang, Z., Ji, R., & Gao, Y. (2019). Hypergraph Neural Networks. AAAI 2019 (message passing on hypergraphs without collapsing to a graph).
• Papillon, M., Sanborn, S., Hajij, M., & Miolane, N. (2023). Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks. arXiv:2304.10031 (a taxonomy of higher-order message-passing models).