Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Differentiable Persistence: Backpropagating Through Topology

3 minute read

Published:

TL;DR: Given filtration values f = (f₁, ..., fₙ) on simplices, the persistence diagram dgm(f) is a piecewise-linear function of f — each birth and death is a specific fᵢ value. The gradient ∂ℒ/∂f of a loss ℒ(dgm(f)) can be computed by chain rule, with the Jacobian of dgm having a sparse structure: each persistence pair (b, d) = (fᵢ, fⱼ) contributes gradients ±1 to positions i and j.

Filtration Values and Diagrams

Fix a simplicial complex \(K\) with \(n\) simplices. A filtration is determined by function values \(f = (f_1, \ldots, f_n) \in \mathbb{R}^n\) on simplices (simplices enter in order of increasing \(f_i\)). The persistence diagram \(\mathrm{dgm}(f)\) consists of birth-death pairs \((f_i, f_j)\) for paired simplices \((\sigma_i, \sigma_j)\).

Key observation: The map \(f \mapsto \mathrm{dgm}(f)\) is piecewise linear. The combinatorial pairing (which simplex pairs with which) is fixed on each “chamber” of \(\mathbb{R}^n\) where the ordering of \(f\) values stays the same. Within a chamber, each diagram point is a linear function of \(f\).

The Gradient Formula

For a loss \(\mathcal{L}(\mathrm{dgm}(f))\) that depends on diagram points \((b_k, d_k)\):

$$\frac{\partial \mathcal{L}}{\partial f_i} = \sum_{k : b_k = f_i} \frac{\partial \mathcal{L}}{\partial b_k} - \sum_{k : d_k = f_i} \frac{\partial \mathcal{L}}{\partial d_k}$$

The \(-\) sign for death comes from the convention that the pairing algorithm sets \(d_k = f_j\) where \(\sigma_j\) is the negative (death) simplex. Moving \(f_j\) up increases the death time, prolonging the feature.

Topological Losses

Several useful losses exploit this gradient:

Total persistence: \(\mathcal{L}_{tp}(f) = \sum_{(b,d) \in \mathrm{dgm}(f)} (d - b)^p\)

Minimising this pushes all features to die quickly (kills spurious topology). Maximising concentrates persistence into a few long-lived features.

Betti-matching loss: \(\mathcal{L}_{bm}(f) = d_B(\mathrm{dgm}(f), \mathrm{dgm}_{\text{target}})^2\)

Minimise the bottleneck (or Wasserstein) distance from a target diagram. Used to force a learned representation to have a prescribed topological structure.

Loop-length loss (for cycles): penalise or reward the birth/death times of \(H_1\) features.

Practical Considerations

  • Gradients are well-defined everywhere except at points where the pairing changes (a measure-zero set).
  • Efficient implementations (TopoLayer, giotto-tda) use the persistence algorithm with automatic differentiation.
  • Backpropagating through \(\mathrm{dgm}\) into image pixels: for a pixel intensity function, each birth/death has a gradient w.r.t. the two “critical” pixels that determined the pair.
Key Insight: The differentiability of persistence enables a new class of regularisers: instead of penalising the norm of weights (L2), you can penalise the topological complexity of learned representations. For example, "enforce that the learned point cloud representation has at most k connected components" becomes a soft constraint via the total persistence loss on H₀. This is a fundamentally new type of inductive bias that has no equivalent in classical regularisation.

Applications

  • Topology-aware autoencoders: encode data so that the latent space has prescribed topology (e.g., a torus for periodic data).
  • Network architecture shaping: train neural networks with topological constraints on weight space connectivity.
  • Medical image segmentation: topological losses ensure that segmented vessels/bronchi are connected (no spurious breaks).

References

  • P. Gabrielsson et al., “A Topology Layer for Machine Learning,” AISTATS 2020. arXiv:1905.12200.
  • N. Nakamura et al., “Topological Loss Term for Biomedical Image Segmentation,” CVPR Workshops 2019.
  • X. Hu et al., “Topology-Preserving Deep Image Segmentation,” NeurIPS 2019.

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