math.AGOct 15, 2026
SaveDeep Learning of Topological Equivalence Classes
Abstract
We demonstrate an application of deep learning to the problem of identifying whether two generating vectors correspond to topologically equivalent finite group actions on Riemann surfaces, showing how architectures developed for semantic understanding have use for pure mathematical problems. In doing so we discuss how desired algebraic structure influences model design, and explore which structural properties of topological equivalence classes the model has successfully learnt.
Figures & tables
Figure 1. The twin network architecture. The model has two identical halves, each half processing one generating vector. A tokenised element of a generating vector is passed through three identical processing blocks (with distinct weights) in series: the output of a convolutional encoder is concatenated with the block input, an alignment layer relates the encodings to those of the corresponding generating vector in the opposite half, and the outputs are fused and pooled over blocks. Each element of the generating vector is processed in parallel through this multiblock structure (aligning with its corresponding element in the opposite generating vector). These vectors are augmented by adding residuals constructed by passing them through a 1d convolution and GeLU activation, and the maximum over elements gives a feature vector for each generating vector as input for a prediction layer, answering the question whether two generating vectors lie in the same topological equivalence class.
| Signature Length | Accuracy | |
|---|---|---|
| Random Embedding | Matrix Embedding | |
Table 1. Model accuracy on test pairs when trained on pairs with unrestricted and varying range for . ∗ indicates that identity padding was used.
| Training Pairs | Accuracy |
|---|---|
Table 2. Accuracy against training set size (train , test )
| Subclass | Accuracy | Number of Pairs |
|---|---|---|
| all pairs | ||
| cross-group | ||
| same-group non-abelian | ||
| same-group cyclic | ||
| same-group other abelian | ||
| same-group -group |
Table 3. Model accuracy on subclasses of generating vector pairs
Figure 2. ROC curves for the model classifying pairs as in the same equivalence class. Subclasses are labelled by the AUC.
Figure 3. ROC curves for the model classifying pairs as the confidence threshold is varied. Subclasses are labelled by the AUC.
Figure 4. Dimensional reduction of the absolute elementwise difference in the generating vector features of a pair, stratified by subclass
Figure 5. Dimensional reduction of the elementwise product of the generating vector features of a pair, stratified by subclass