Learn the readout instead of hand-picking it — topological structure as input to a model, or as the domain the model operates on.

Topological Deep Learning

Two things under one name

“Topological deep learning” covers two distinguishable practices, and papers rarely separate them. The distinction is worth holding because they fail differently.

Topology as features. Compute a topological summary — a barcode, a persistence image, a Laplacian spectrum — vectorize it, and feed it to a conventional model. The topology is a fixed preprocessing step; the network never sees the complex. Most molecular work in this corpus is this kind, and it is the older and better-validated of the two.

Topology as domain. Build the network to operate on the complex itself, with message passing over simplices or cells rather than over graph edges. Here the topological structure is where computation happens, not what is handed to it. simplicial-learning is this branch.

The first is an engineering choice about representation. The second is an architectural claim about what a network should be. Conflating them makes the literature’s evidence look stronger than it is, because results from the well-trodden first practice get read as support for the second.

Why it displaced hand-picked summaries

A barcode is not a vector, and everything downstream needs one. The bridge is a vectorization — persistence images, landscapes, Betti curves, binned statistics — and every one of them embeds a choice about which parts of the diagram matter. Choosing by benchmark is choosing a hyperparameter on your test set.

Letting a model learn the readout is the natural response, and it is why the field moved. The cost is that the guarantees do not survive the move: persistent homology’s stability theorem bounds how much the diagram moves, and says nothing about how much a learned function of it moves. Stability is a property of the summary, not of the pipeline.

The pattern that recurs

Across the applications in this corpus the successful shape is consistent, and it is worth stating because it is easy to get backwards:

Restrict to the region that matters — a binding interface, a pocket, a neighbourhood — rather than computing over the whole structure.
Condition the complex on chemistry. Separate filtrations per element type or per interaction class, so the summary cannot average away the distinction the task depends on.
Summarize topologically.
Learn on the result, usually alongside non-topological features.

Step 2 is where most of the gain lives. Plain persistent homology on all atoms at once is invariant to exactly the chemical differences that determine binding, and no amount of model capacity downstream recovers information the representation destroyed. The surveys in this corpus both make this their central methodological point: successful molecular TDA is conditioned topology.

Step 4 is where most of the overclaiming lives. When a topological model beats a baseline, the question is whether the topology contributed or the auxiliary features and the training setup did. An ablation that removes the topological block is the cheapest honest answer, and it is frequently absent.

How to read results in this area

The benchmarks are heterogeneous and the reported numbers are often not comparable. Three habits help:

Check the split. Random splits on structural data leak, because homologous structures land on both sides. Family-, scaffold-, or time-aware splits are the meaningful ones.
Check whether correlations are pooled or per-target. A single correlation computed across all targets at once can look strong while per-target ranking — the thing a user actually needs — is weak.
Check what the baseline is. “Outperforms persistent homology” and “outperforms the current best structure predictor” are very different claims.

This corpus has direct experience of the third and second points: replicating a published topological scorer reproduced its numbers while establishing that part of its reported margin came from an implementation defect rather than from the method.

What implements it here

topoqa — graph attention over interface residues with topological node features, for protein complex interface quality assessment.
topodockq — persistent-Laplacian features for peptide–protein complex assessment.
hiponet — the domain branch: higher-order networks over simplicial structure.
Fixtures: topodockq-environment, hiponet-environment, open-topoqa-scorer-environment.

persistent-homology and persistent-laplacian are the usual feature sources. simplicial-learning is the architectural branch. Both survey notes — tda-tdl-molecular-sciences and tda-tdl-beyond-persistent-homology — are surveys of this area specifically and are the right starting point for the literature.