Topology-Preserving Distance Alignment

• Topology-preserving distance alignment is a computational framework that rigorously maintains both local and global topological structures in data representations.
• Methodologies include persistent homology, bottleneck distances, and autoencoder optimizations to align geometric and topological invariants across embedding spaces.
• Applications span dimensionality reduction, federated learning, and multilingual embedding alignment, enhancing model robustness and interpretability.

Distance alignment for topology-preserving representations refers to the rigorous computational and mathematical methodologies that enforce the preservation and alignment of distances and associated global and local topological features within data representations. This principle is central to a range of tasks in representation learning, dimensionality reduction, federated learning, and multimodal/multilingual embedding alignment, where the preservation of topological invariants—such as clustering structure, connectivity, loops, and higher dimensional cycles—between source and target spaces or across distributed agents is critical. Distance alignment mechanisms exploit persistent homology, interleaving metrics, bottleneck distances, and auxiliary graph-based distances to guarantee or improve the faithfulness of the learned or aligned representations to the intrinsic topology of the original data distribution or reference space.

1. Mathematical Foundations: Interleaving Distance and Bottleneck Metrics

A fundamental mathematical framework for topology-preserving distance alignment is built from persistent homology and functorial representations of filtrations as persistence modules. Key constructs include the interleaving distance $d_I$ between persistence modules and the bottleneck distance $d_B$ between associated barcodes or persistence diagrams. For representations $M,N: P \to K$-mod over a finite poset $P$ endowed with cover-relation lengths, one defines order-preserving translations $\Lambda: P^+ \to P^+$ with heights $h(\Lambda)$ and natural transformations as the components of a $(\Gamma,\Lambda)$-interleaving. The infimal $\epsilon$ for which an interleaving exists yields $d_I(M,N)$. The module decompositions over $P^+$ induce barcodes $B(M), B(N)$, and the bottleneck distance is defined via $\delta$-matchings of indecomposable intervals with controlled widths. The classical isometry $d_I(M,N) = d_B(M,N)$ holds under regularity conditions, ensuring that distance alignment at the algebraic module level translates directly to topological alignment at the representational level (Meehan et al., 2017).

Discretization schemes, such as those leveraging finite meshes $X$ approximating real-valued filtration scales, make these distances computationally tractable. Theorems guarantee convergence of the discretized interleaving and barcode metrics to their continuous counterparts as the mesh refines, providing a practical pathway to topology-preserving distance alignment in data analysis.

2. Autoencoder-Based Approaches for Topology Preservation

Topology-preserving autoencoders explicitly regularize or optimize their embeddings to align either local or global geometric and topological structures. Multiple methods exemplify this paradigm:

• Manifold-Matching Autoencoder (MMAE): Introduces an unsupervised regularization term that directly penalizes the mean squared error between pairwise distances in the latent and input (or preprocessed) spaces. On a minibatch of $b$ points, MMAE minimizes

$$\mathcal{L}_{\mathrm{MMAE}} = \frac{1}{b}\sum_i \|x_i - g_d(f_e(x_i))\|^2 + \lambda \frac{1}{b^2} \sum_{i,j} (\|z_i - z_j\| - \|e_i - e_j\|)^2$$

where $z_i=f_e(x_i)$. By aligning all pairwise distances, this approach guarantees, via the Gromov–Hausdorff stability of Vietoris–Rips persistence, that the persistent homology (and thus the topological features) of the latent code surface closely matches that of the input data. MMAE is scalable and provides a parametric, out-of-sample extension of classical MDS (Cheret et al., 17 Mar 2026).

• Representation Topology Divergence Autoencoder (RTD-AE): Minimizes the RTD—defined using cross-barcode construction on a doubled graph over input and latent points—within the training objective

$$\mathcal L(\theta, \phi) = \frac{1}{2}\|X_{\mathrm{batch}} - X_{\mathrm{rec}}\|^2_F + \lambda\, \mathrm{RTD}(X_{\mathrm{batch}}, Z_{\mathrm{batch}})$$

RTD-AE provides theoretical guarantees: RTD$=0$ implies isomorphism of persistence barcodes including feature locations, and the method comes with Lipschitz stability wrt distance matrices. Empirical results confirm superior preservation of both local and global clustering, manifold loops, and distributions of topological event scales (Trofimov et al., 2023).

• Interleaving Optimization: Approaches such as dimensionality reduction via direct minimization of the interleaving/bottleneck distance between the persistent homology of the data and its embedding provide an interpretable mechanism to enforce topological faithfulness. Explicit gradients and efficient subroutines allow backpropagation through the persistent homology pipeline (Nelson et al., 2022).

These methods align distances not only to preserve local neighborhoods but also complex topological structures, such as multi-component, high-genus, or looped manifolds.

3. Topology-Informed Distance Alignment in Federated and Distributed Representations

In federated learning and decentralized representation learning, heterogeneous (non-IID) client data can cause divergence and topological inconsistency in learned features. Distance alignment for topology-preservation in these contexts leverages persistence-based regularization:

• FedTopo: Implements a three-stage process: (a) Topology-Guided Block Screening (TGBS) selects the network layer with maximal topological separability via persistence-based ROC-AUC; (b) Topological Embedding (TE) computes channel-averaged Persistence Images from persistence diagrams of activations, providing a robust, globally stable descriptor; (c) Topological Alignment Loss (TAL) penalizes squared Euclidean distance between client and global TEs. This framework provably accelerates convergence and improves accuracy under severe data heterogeneity, outperforming a broad suite of federated and non-federated benchmarks (Hu et al., 16 Nov 2025).

The practical recipe includes server-side block screening, distributed computation of TE on each client, and convergence monitoring of both classification loss and TE consistency. Empirical results demonstrate that this persistent-homological alignment enforces topological coherence and reduces representation drift under federation.

4. Distance- and Topology-Alignment in Multi-View and Multilingual Models

Contrastive learning frameworks, such as vision-LLMs (VLMs) and multilingual embedder distillation, have evolved to recognize that instance-level alignment is insufficient for global structural correspondence:

• Contrastive Alignment on Oblique Manifolds: The softmax temperature in standard visual-textual alignment (e.g., CLIP) is interpreted as a scaling parameter for the induced distance distribution, modulating the effective topology of the embedding space. Alternative architectures introduce an oblique manifold topology for embeddings, representing each as a matrix with column-normalization, and employ the negative inner product as a distance function (removing the need for large temperature parameters). This design yields more favorable range, stability, and interpretability in alignment, and leads to marked improvements (+6% zero-shot accuracy) over standard CLIP models (Sun, 2022).
• Topology Alignment in Multilingual CLIP (ToMCLIP): Augments textual distillation objectives with (i) a persistent-homology-derived topological alignment loss (sliced Wasserstein between persistence diagrams of teacher and student embeddings) and (ii) a pairwise Euclidean distance-matrix alignment loss. The adoption of persistence-based losses, with computational surrogates via MST-based graph sparsification, ensures that global geometric features of the reference (e.g., English) embedding manifold are preserved in the multilingual embedding space. ToMCLIP demonstrates empirical gains in cross-lingual retrieval and robustness, confirming the necessity of topological coherence in the transfer process (You et al., 13 Oct 2025).

5. Multi-Field Topological Distance Measures and Structured Data Analysis

For applications involving multi-field or spatiotemporal data, topological alignment requires hierarchically organized descriptors:

• Multi-Dimensional Reeb Graphs (MDRG): The MDRG construction provides a hierarchy of Reeb graphs for multi-field data, from which one computes a set of persistence diagrams (e.g., ordinary, superlevel, and extended persistence in various homological dimensions). The extended bottleneck distance $d_T$ between two MDRGs extends the bottleneck pseudometric to multi-field contexts and incorporates combinatorial bijections between same-level Reeb graphs. Stability bounds guarantee Lipschitz continuity under input perturbations. This structure enables robust and theoretically sound shape classification and dynamic event detection (e.g., in time-varying molecular chemistry) (Ramamurthi et al., 2023).

These hierarchically defined distances operationalize alignment between multi-view structures and guarantee that only features of similar persistence and dimension are matched, thus preventing collapse or dilution of essential topological signatures.

6. Theoretical Guarantees and Stability Results

Topology-preserving distance alignment methods are underpinned by stability theorems from persistent homology. The bottleneck distance between persistence diagrams is bounded by twice the Gromov–Hausdorff distance between metric spaces (Cohen-Steiner et al., 2007), and uniform control of pairwise distance distortion suffices to guarantee the preservation of all topological features above prescribed scales. Specific metrics, such as RTD or $d_T$ for MDRGs, possess Lipschitz bounds and pseudo-metric properties, ensuring that small errors in geometric alignment do not cause catastrophic topological errors. Convergence results show that discretized poset-based computations or sparse graph surrogates recover true barcode distances in the limit (Meehan et al., 2017, You et al., 13 Oct 2025, Trofimov et al., 2023, Ramamurthi et al., 2023).

7. Empirical Benchmarks and Comparative Results

Empirical studies across synthetic and real-world domains consistently confirm that topology-preserving distance alignment methods yield improvements on metrics sensitive to geometric and topological faithfulness—such as Wasserstein distances between persistence barcodes ($W_0$), linear distance correlation (DC), triplet accuracy (TA), nearest-neighbor preservation, and cluster density recovery. Benchmarks include nested spheres, linked tori, high-dimensional image datasets, single-cell RNA sequencing, and multilingual retrieval. Methods such as MMAE, RTD-AE, and FedTopo outperform previous baselines on both geometric and topological metrics without sacrificing scalability or differentiability (Cheret et al., 17 Mar 2026, Trofimov et al., 2023, Hu et al., 16 Nov 2025).

---

Distance alignment for topology-preserving representations constitutes a rigorously founded and empirically validated paradigm that unifies persistent homology, metric geometry, and machine learning. This paradigm extends from single-device learning to federated, multimodal, and multi-field contexts, and is distinguished by its dual capability to enforce both local geometric fidelity and preservation of critical global topological invariants. Recent work provides both foundational theorems and scalable algorithmic recipes, establishing distance alignment as a key principle underlying robust, interpretable, and theoretically sound representation learning across data modalities and architectures.