Topological Alignment Loss (TAL)

• Topological Alignment Loss (TAL) is a family of loss functions that penalizes discrepancies in global topology by comparing persistence diagrams using metrics like Wasserstein distances.
• Different TAL variants, such as morphological closing-based for 2D segmentation and Wasserstein-based for 3D and graph data, adapt to various data types and applications.
• TAL integrates with standard network losses to guide training toward structural consistency, showing measurable gains in biomedical imaging, connectomics, and federated learning tasks.

Topological Alignment Loss (TAL) is a family of loss functions designed to enforce topological consistency between model predictions and targets, or between representations in different models or modalities, by explicitly penalizing topological discrepancies. These losses operationalize the principles of persistent homology, morphological analysis, and optimal transport on persistence diagrams or barcodes to guide learning toward global structural fidelity, rather than merely matching predictions at a pixel, voxel, or pointwise level.

1. Mathematical Formulation and Core Variants

TAL encompasses multiple formalizations, each tailored to the data type (images, volumes, graphs, manifolds, etc.) and topological signals of interest. However, the common thread is the quantification and minimization of a global topological distance between two objects—typically via persistence diagrams in homologies of dimension 0 (connected components), 1 (loops), and higher.

a) Morphological Closing-Based TAL for 2D Vascular Segmentation

The original "Topological Alignment Loss" in blood vessel segmentation is defined by leveraging the morphological closing operator to penalize gaps and spurious bridges in network predictions. Let $\mathbf{Y}$ be the binary ground-truth mask and $\mathbf{P}$ the predicted probability mask. For each radius $r$:

• Apply dilation and erosion (closing) using a square structuring element $S_r$ to identify which gaps would be “bridged.”
• Compute the per-radius error map for missing branches:

$$\epsilon_r\,(\mathbf{P}_{C(r)} - \mathbf{P})^2\,\cdot\,\mathbf{Y}_s$$

where $\mathbf{Y}_s$ is the ground-truth vessel skeleton.

• The total “missing-branch” error is:

$$E_\mathrm{miss}(\mathbf{P},\mathbf{Y}) = \sum_{r=1}^{r_M} \epsilon_r(\mathbf{P}_{C(r)}-\mathbf{P})^2\,\mathbf{Y}_s$$

• Two normalized terms: topology-sensitivity (missing branches) and topology-precision (false bridges), combined with a trade-off $\alpha$:

$$L_\mathrm{topo}(\mathbf{P},\mathbf{Y};\alpha,r_M) = \alpha L_\mathrm{tsens} + (1-\alpha)L_\mathrm{tprec}$$

This composite loss is integrated with standard pixelwise networks losses (Araújo et al., 2021).

b) Wasserstein Distance-Based TAL on Persistence Diagrams

Persistent-homology-based TALs align the topology of prediction and target by minimizing the $p$-Wasserstein or sliced-Wasserstein distance between their persistence diagrams. Given diagrams $D, D'$, their $p$-Wasserstein distance is:

$$W_p(D, D') = \left(\inf_{\eta} \sum_{x\in D} \|x - \eta(x)\|^p_\infty \right)^{1/p}$$

where $\eta$ is a bijection extended with diagonals. In 3D segmentation, the total loss adds a per-voxel geometric term (e.g., Dice) and a multi-dimensional topological term summing over $q=0,1,2$ (connected components, tunnels, voids):

$$L_\mathrm{TAL}(f_\mathrm{gt}, f_\mathrm{pred}) = L_\mathrm{geo}(f_\mathrm{pred},f_\mathrm{gt}) + \lambda \sum_{q=0}^2 \left[ W_2(\mathrm{PD}_q(f_\mathrm{gt}), \mathrm{PD}_q(f_\mathrm{pred})) + \mathrm{Pers}_2(\mathrm{PD}_q(f_\mathrm{pred})) \right]$$

This approach is end-to-end differentiable and widely used in shape reconstruction and singular-object segmentation (Waibel et al., 2022, Wen et al., 3 Dec 2024).

c) Graph and Network Barcodes

For weighted graphs $G=(V,w), H=(V,w')$, the barcodes $I_0$ (0D, MST births) and $I_1$ (1D, non-MST edges) yield vectorized summaries. The TAL is the sum of squared-matching costs:

$$L_\mathrm{top}(G,H) = \sum_{i=1}^{m_0}(b_i-b'_i)^2 + \sum_{j=1}^{m_1}(d_j-d'_j)^2$$

Here, $b$ and $b'$ are sorted edge weights corresponding to birth times, and $d,d'$ are death times. This reduces combinatorial complexity and captures both connectivity and cycles (Songdechakraiwut et al., 2020).

d) Feature-Space and Embedding Alignment

Some applications extract compact topological embeddings (via persistence images), and align those directly in feature space with a squared-Euclidean penalty:

$$\mathcal{L}_\mathrm{TAL}(w_i,\bar w) = \mathbb{E}_{x \sim \mathcal{D}_i} \| \mathbf{t}_i(x;w_i) - \mathbf{t}_i(x;\bar w)\|_2^2$$

where $\mathbf{t}_i$ is a summary of persistent features from activations at a chosen network block (Hu et al., 16 Nov 2025).

2. Persistent Homology and Topological Features

At the core of all TAL variants is the use of persistent homology to quantify multi-scale topological features.

• Filtration: A nested sequence (e.g., of thresholded images, cubical or simplicial complexes).
• Homology: $H_k$ tracks $k$-dimensional features—$H_0$ (components), $H_1$ (cycles), $H_2$ (voids).
• Persistence diagrams: Each tuple $(b,d)$ records the birth and death threshold of a topological feature; the set forms the diagram or barcode.
• Wasserstein metrics: Distances (e.g., $W_2$) provide stable, globally aware measures of topological similarity.

Some methods enrich alignment by incorporating spatial coordinates of feature "creators" to mitigate ambiguous matching (Wen et al., 3 Dec 2024).

3. Training Integration and Computational Aspects

TAL is typically applied as a regularizer:

• Segmentation: Combined with Dice or cross-entropy loss, morphological operations (closing) are efficiently implemented via GPU max/min pooling; computational overhead is $O(BHW r_M)$ for closing-based TAL (Araújo et al., 2021).
• Persistent homology calculation: For images and volumes, cubical or simplicial complexes are constructed and persistence extracted using libraries like GUDHI or Ripser; in practice, diagram computation is tractable after downsampling or by focusing on large-persistence features (Waibel et al., 2022).
• Graph-based methods: Barcodes (births and deaths) are extracted from the MST and non-MST edges, with overall $O(|E| \log |V|)$ complexity (Songdechakraiwut et al., 2020).
• Sliced Wasserstein approximation: Batched point clouds (e.g., CLIP embeddings) use projected one-dimensional Wasserstein computations for differentiability and scale (You et al., 13 Oct 2025).
• Federated/representation learning: Topological descriptors are extracted per-sample, per-block, then compared via squared-Euclidean or Wasserstein penalties; block selection is guided by precomputed topological “separability” (Hu et al., 16 Nov 2025).

4. Empirical Results and Application Domains

TAL shows consistent gains across diverse domains:

Domain	Task	TAL Variant	Quantitative Gains
Vessel segmentation	2D mask topology	Morph. closing-based	TSI scores $\sim$+0.5–3% / improved branch connectivity (Araújo et al., 2021)
3D biomedical recon	Cell and nuclei shapes	PD Wasserstein	IoU error $\downarrow$ 0.49$\to$0.47; volume and surface error drops (Waibel et al., 2022)
Brain connectomics	Functional/structural graphs	Barcode $L_2$-TAL	More heritable edges detected; strong group discrimination (Songdechakraiwut et al., 2020)
Federated learning	Client–global alignment	TE–Euclidean-TAL	+13–16% accuracy under severe non-IID (Hu et al., 16 Nov 2025)
Multilingual CLIP	Vision-language embeddings	SW2 diagram loss	+0.8–1.3% zero-shot top-10 accuracy, improved topological metrics (You et al., 13 Oct 2025)

Empirical evaluations demonstrate that adding TAL systematically improves global structure preservation, especially in scenarios where topology is not captured by pixelwise or instance-wise losses.

5. Methodological and Implementation Considerations

• Choice of Homology Degree: $H_0$ (components) is fastest and usually critical; $H_1$ (loops) may be required for cycles/holes; $H_2$ (voids) for volumes.
• Differentiability: Morphological and graph-based TALs exploit subdifferentiable pooling or treat persistence pairings as fixed during infnitesimal parameter updates.
• Spatial-Awareness: Ambiguities in diagram matching are mitigated by including spatial proximity weights in assignment costs (Wen et al., 3 Dec 2024).
• Computational Scaling: GPU implementations of morphological operations, diagram downsampling, and sliced or entropic-regularized optimal transport mitigate cost for large-scale data.

6. Generalization, Strengths, and Limitations

TAL is highly general and has been adapted for:

• Medically critical segmentation (vascular, bronchial, neural).
• 3D object/biomedical structure reconstruction.
• Functional brain network alignment across modalities/subjects.
• Federated and distributed representation-learning under non-IID data.
• Multimodal multilingual model alignment.

Notable limitations include sensitivity to hyperparameters (e.g., closing radius, $\alpha$ trade-off, diagram truncation), possible over-connection if input skeletons are noisy, and increased compute cost for higher-dimensional homology or large diagrams. Some methods may slightly over-estimate thin structure thickness when enforcing topological coherence (Araújo et al., 2021).

7. Relationship to Other Topological and Geometric Losses

TAL differs from early topological losses using only Betti numbers or simple connected component counts by leveraging persistence-based assignments, spatial-aware or barcode-weighted matching, and, in some cases, explicit morphological analysis. Compared to pairwise-matching or optimal transport methods not informed by topology, TAL provides theoretically justified, globally meaningful gradients that respect both topological and, when extended, spatial constraints (Waibel et al., 2022, Wen et al., 3 Dec 2024). In representation learning, TAL complements instance-level or geometrically local constraints, yielding improved cross-modal, cross-client, or cross-language alignment (Hu et al., 16 Nov 2025, You et al., 13 Oct 2025).