Topological Deep Learning

Generalized morphology control requires a single policy to transform information across limbs with different physical roles, coordinate whole-body motion, and remain efficient as body size grows. Existing communication mechanisms address these requirements only partially. We introduce RecMorph, a topology-guided spatial recurrent architecture that uses recurrent sequence computation to jointly perform cross-limb communication and representation transformation. A depth-first traversal converts the kinematic tree into a morphology-derived sequence, along which shared bidirectional transitions progressively transform limb information before action decoding. Residual preservation, RMS normalization, and input-dependent channel modulation stabilize this repeated spatial transformation, yielding linear token complexity at fixed model width and depth. Across five UNIMAL tasks, RecMorph achieves the strongest mean final training performance among the evaluated generalized morphology controllers and the highest measured inference throughput on FT, while generalizing to unseen variations and bodies with up to 30 limbs. We further migrate representative generalized controllers from UNIMAL benchmarks to a four-platform quadruped setting. RecMorph achieves the best macro-averaged performance under nominal and high friction, reduces nominal velocity RMSE by 43.5% relative to specialist MLPs, and one shared policy completes 40 physical Go1/Go2 trials without falls. These results show that topology-guided recurrent transformation provides an effective and efficient communication mechanism for Generalized Morphology Control and remains effective when transferred from procedural bodies to physical robot platforms. Code and experimental resources are publicly available at https://github.com/quanruirao/RecMorph.

Edge vision models are difficult to deploy on resource-constrained hardware, making low-bit post-training quantization (PTQ) attractive. In practice, standard FP32 training often produces heavy-tailed activation distributions whose outliers destabilize activation quantization: preserving the full range wastes quantization bins on rare extremes, while aggressive clipping causes information loss. Existing solutions typically rely on quantization-aware training (QAT), which adds training complexity and bit-width coupling, or advanced PTQ procedures that repair the model after training. We present SCULPT (Statistical Clipping and Uniform Loss for Post-Training), a training-time method that improves PTQ readiness during ordinary FP32 fine-tuning. SCULPT combines a topology-aware activation regularizer that suppresses quantization-hostile skewness and kurtosis with a stable percentile-based clipping mechanism that learns deployment-ready activation bounds. Unlike QAT, SCULPT does not simulate quantization during optimization; unlike post hoc outlier-repair PTQ methods, it does not require runtime activation transformations. The learned clipping bounds can be exported directly into a standard PTQ workflow for low-bit deployment, including INT8 and lower-bit settings such as W4A8.

Voucher abuse poses a major challenge in e-commerce, where malicious users exploit promotional vouchers for profit. Unfortunately, fraud patterns evolve rapidly over time and across regions, causing distribution shifts that degrade existing detection models unless retrained frequently. To tackle this, we propose the Coupled Attribute-Topology Invariance Learning framework (CATeye). The key challenge arises from coupled attribute-topology shift, where edges built from attribute proximity cause environment-driven attribute shift to induce shifted topology, thereby amplifying variant signals through GNN message passing. CATeye sees through such coupled shifts with two learnable selectors. First, an Attribute Invariance Selector (AIS) learns node-adaptive masks to filter out non-invariant attributes. Then, conditioned on retained invariant attributes, an Edge Invariance Selector (EIS) samples an invariant subgraph and isolates non-invariant edges. Using the resulting invariant and non-invariant components, CATeye constructs multiple views and applies view-specific objectives to emphasize domain-invariant representations while suppressing domain-specific variations. Experiments on both a proprietary dataset from Lazada, a major Southeast Asian e-commerce platform, and a public benchmark show that CATeye consistently outperforms nine strong domain generalization and graph anomaly detection baselines, achieving up to an 8.61% improvement in average F1 score over the strongest baseline. Source code is publicly available at https://github.com/Tian0426/CATeye.

Domain adaptation has advanced cross-scene hyperspectral image classification, significantly improving discriminative capability in complex scenarios. However, privacy rules or storage limits often block access to data from the source domain. Conventional domain adaptation methods become impractical, severely restricting their utility in realistic remote sensing scenarios. To tackle this challenge, we propose a topology-aware source-free learning framework. We first introduce the entropy momentum pseudo-labeling (EMP) to refine k-means assignments by leveraging entropy-aware confidence and temporal prediction momentum. Under the guidance of the refined pseudo-labels, we further utilize the contextual neighborhood topology (CNT) to exploit the intrinsic geometric structure of the target feature space. Combining the global structural information extracted by collaborative representation with the local similarity information modeled by nearest neighbor search, the CNT accomplishes the comprehensive encoding of manifold-level geometric properties in the target domain feature space. The overall objective integrates cross-entropy on refined pseudo-labels, log inner product-based topology consistency, and an information-maximization term for balanced classification, ensuring stable adaptation in the source-free setting. Extensive experiments on three typical cross-scenarios demonstrate that the proposed method exceeds state-of-the-art performance, and ablation studies further validate the contribution of each module. The results highlight the critical role of topology-aware modeling in achieving robust and accurate classification without source data.

Contemporary model-free reinforcement learning algorithms can achieve very high performance, but have low sample efficiency and are not robust to changes in the environment. Model-based algorithms have much higher sample efficiency, but still fail when the environment shifts. This paper introduces Adaptive Topological Learning with Abstract Successors (ATLAS) to combat these challenges. ATLAS uses a Grow When Required network with Successor Features in order to achieve high sample efficiency while also robustly tackling catastrophic forgetting. We evaluate ATLAS in spatial navigation tasks, benchmarking its performance against common on-policy and off-policy algorithms. Our empirical results demonstrate that by structurally decoupling transition dynamics from the reward signal, ATLAS achieves near-instantaneous adaptation to new goals and can exhibit positive backward transfer, significantly outperforming baseline methods in non-stationary environments.

Road network segmentation from satellite imagery remains challenging due to large geographic variation in road appearance, occlusions, and domain shifts introduced by differing resolutions and sensors. Existing models, typically trained under narrow resolution--region combinations, generalise poorly to unseen environments such as rural settings, regions with distinct road materials, or imagery from new satellite platforms, often producing broken or disconnected predictions. Adapting these models to new domains usually requires retraining or fine-tuning, which is costly and risks catastrophic forgetting. In this work, we reframe global road extraction as a continual adaptation problem rather than an architectural one. Our framework combines cross-resolution knowledge distillation across a resolution-decreasing curriculum, multi-sensor training, and topology-aware supervision, yielding a single model that generalises across $0.3-1.0$ m imagery from multiple satellite platforms across continents. On publicly available benchmarks, including City-Scale and Global-Scale, our model outperforms state-of-the-art results by up to $22$ F1 points and $15$ APLS points, while remaining the most efficient, with $3\times$ faster inference. Our results suggest that improved robustness across diverse sub-meter satellite imagery can be achieved through targeted training strategies, such as data curricula, distillation, and topology-aware losses, rather than increasingly complex architectures.

Topological neural networks (TNNs) enable leveraging high-order structures on graphs (e.g., cycles and cliques) to boost the expressive power of message-passing neural networks. In turn, however, these structures are typically identified a priori through an unsupervised graph lifting operation. Notwithstanding, this choice is crucial and may have a drastic impact on a TNN's performance on downstream tasks. To circumvent this issue, we propose $\partial$lift (DiffLift), a general framework for learning graph liftings to hypergraphs and cellular- and simplicial complexes in an end-to-end fashion. In particular, our approach leverages learned vertex-level latent representations to identify and parameterize distributions over candidate higher-order cells for inclusion. This results in a scalable model which can be readily integrated into any TNN. Our experiments show that $\partial$lift outperforms existing lifting methods on multiple benchmarks for graph and node classification across different TNN architectures. Notably, our approach leads to gains of up to 45% over static liftings, including both connectivity- and feature-based ones.

Federated learning (FL) enables collaborative learning over decentralized data silos without centralizing raw data. However, heterogeneous local architectures often induce non-aligned representation spaces, making it difficult to transfer global knowledge across silos. Existing paradigms share this knowledge as model parameters, distilled predictions, or class prototypes, yet all encode it in an absolute space that must be aligned across clients. Heterogeneous backbones break this alignment, so the shared knowledge becomes unreliable and misleads local training. We propose FedTopo, a relation-level framework that encodes global knowledge as class relation topology, capturing how classes relate within each client rather than where they lie in feature space. Each client builds its relation topology from local prototypes and uploads it with class statistics. The server then aggregates these relations in a reliability-aware manner that down-weights weakly supported ones, and broadcasts the global topology to clients. The global topology guides local training by emphasizing topology-similar negative classes. Experiments on three datasets under eight heterogeneous backbones show that FedTopo consistently outperforms parameter-, distillation-, and prototype-sharing baselines, with low communication and no inference overhead. Our code is available at https://github.com/Zhaoyang-Ma/FedTopo.

Automatic feature engineering (AutoFE) for tabular data requires discovering informative transformations from a large program space. Existing approaches suffer from three limitations: classical methods rely on fixed operator libraries with limited expressivity, LLM-based methods generate proposals from static prompts without retaining search experience, and evolutionary methods use fixed migration policies that ignore task-specific cross-family transfer utility. We introduce TOPOFE, a framework that formulates AutoFE as graph-structured multi-island evolutionary program search. The transformation space is partitioned into semantically coherent families, each explored by an island through LLM-guided mutation and crossover. Each island maintains a Prompt Adaptation Memory that accumulates accept/reject feedback to steer proposals toward productive regions without parameter updates. To coordinate global exploration, TOPOFE dynamically learns a directed topology graph whose edge weights encode transfer utility between transformation families. Cross-island transfer is triggered by adaptive saturation detection and performed through LLM-mediated hybrid synthesis, enabling discovery of compositional feature programs that cannot emerge from isolated local search. Experiments on 29 tabular datasets show that TOPOFE consistently outperforms most state-of-the-art AutoFE methods on classification and regression tasks. Beyond predictive performance, TOPOFE produces feature sets with lower redundancy and higher representational coverage, while the learned topology graph acquires meaningful task-specific transfer structure correlated with downstream gains. The discovered feature programs transfer reliably across diverse predictors and LLM backbones, demonstrating that improvements arise from TOPOFE's structured search and adaptive coordination rather than backbone-specific generation capability.

Topological maps are key outputs of autonomous driving perception systems, delivering essential road information for path planning. They identify instances such as centerlines and traffic signs, along with their connectivity relationships. Due to the lack of explicit markings for centerlines in real-world environments, the detection of centerline instances remains a significant challenge. To tackle this problem, we propose HGeo-TopoMap, which leverages an explicit prior map and implicit spatial relations to hierarchically boost topological mapping. First, a geometric adaptive learning module is designed for the road structure map obtained via inverse perspective mapping. This module discretely encodes semantic and spatial features from the map, followed by a prior-mask attention mechanism that selectively focuses on informative regions. Then, a geometric consistency learning module is devised, which leverages the geometric properties and spatial relationships of centerlines. Built on the geometry-aware decoder, it enforces spatial consistency by aligning features of centerline instances with identical geometric orientations. The proposed method is evaluated on the OpenLane-V2 dataset across the centerline, lane segment, and robustness benchmarks. Beyond substantial improvements in topological mapping accuracy, the proposed method offers the benefit of enhanced robustness, consistently outperforming baselines under both standard and challenging conditions. The source code and model weights will be made publicly available at https://github.com/lynn-yu/HGeo-TopoMap.

Computer aided design (CAD) is ubiquitous: virtually any modern object was designed using editable CAD tools. However, with the shortage of available CAD datasets in its native editable and parametric format, boundary representation (B-Rep), it is ever more important to develop data-efficient methods for this domain. We present a new self-supervised pretraining task, Masked Topology Modeling (MTM), that leverages the face-adjacency graph, an induced structure unique to B-reps that the encoder can be asked to reconstruct. MTM masks a fraction of edges and trains a small head to predict each masked edge's convexity and curve type from the encoder's post-message-passing face features. We combine MTM with a MoCo-style momentum-queue contrastive learning over B-rep-aware augmentations, a BFS-connected face-region masked-reconstruction objective, and pretraining on the ABC dataset and our new procedurally generated dataset to show strong performance on a number of benchmarks.

Sleep physiology arises from the coordinated dynamics of the central nervous system (CNS) and autonomic nervous system (ANS), as reflected by multimodal polysomnography signals including EEG, EOG, EMG, ECG, and respiration. However, existing sleep foundation models often fuse heterogeneous biosignals in a topology-agnostic manner, overlooking their physiological organization. We introduce Omni-Sleep, a sleep foundation model that uses the CNS/ANS partition as a physiological prior for topology-constrained representation learning. Omni-Sleep learns structured representations through three objectives: intra-system consistency, which captures shared subsystem-level factors within neural and cardio-respiratory signals; inter-system synchronization, which aligns subsystem trajectories to model brain--body dynamics; and latent-space masked temporal modeling, which captures long-horizon sleep dynamics. Pre-trained on over 100,000 hours of multi-center multimodal PSG data, Omni-Sleep is evaluated on sleep staging and multi-disease classification. Across datasets and modality-ablation settings, Omni-Sleep outperforms strong foundation-model baselines, showing improved label efficiency, cross-dataset generalization, and robustness to missing modalities. These results highlight the value of physiological hierarchy for generalizable sleep representation learning. Code is available at https://github.com/AutoBrain-sleep/OmniSleep.

Topological data analysis (TDA), particularly persistent homology (PH), captures geometric structural properties in medical images (e.g., connected components, loops, shape characteristics), which conventional pixel-level deep learning approaches often neglect. While many topological descriptors are known for converting persistence diagrams (PDs) or raw images into topological feature vectors, existing methods mostly default to a single fixed descriptor (e.g., persistence images), leaving the diversity of topological representations largely unexplored. To the best of our knowledge, there is no known large language model (LLM)-based agentic framework that can automatically determine the most suitable topological descriptors for a given image dataset and produce the corresponding topological feature vectors for downstream tasks. To fill this gap, we propose \textbf{TopoAgent}, an LLM-based agentic framework that automates topology learning for medical image analysis.TopoAgent operates through a Perception--Reasoning--Action--Reflection loop supported by 21 domain-specific tools and dual memory that accumulates experience across runs. Its skill set is distilled from systematic evaluation of 15 topological descriptors across 26 datasets with six classifiers. TopoAgent analyzes input images and their topological characteristics, reasons about which topological descriptors best suit the input, and determines the optimal descriptor and its configuration, all without task-specific training.

Floods frequently impact regions around the world. Rapid and accurate flood detection is crucial for emergency response and timely mitigation of human and economic loss. The expanding availability of satellite data and advances in artificial intelligence have enhanced monitoring of environmental hazards, but many flood events remain challenging to detect because cloud cover obscures optical satellite imagery. Rambour et al. introduced the SEN12-FLOOD dataset and extracted per-image features using a ResNet-50 convolutional neural network backbone, then fed these features into a gated recurrent unit network to show that temporal information can substantially improve accuracy compared to single-image baselines. More recently, Chamatidis et al. showed that a vision transformer can achieve strong performance with popular convolutional architectures. However, these models typically function as opaque black boxes, making it difficult to interpret their decision boundaries, learned features, and internal reasoning, especially in safety-critical domains like remote sensing. In contrast, topological data analysis (TDA) provides a mathematically grounded framework for capturing global structural features of data. TDA has emerged as a powerful tool for analyzing complex imagery, especially imagery with geometrically interpretable structures, of which floods are a prime candidate. In this work, we systematically evaluate topological descriptors for flood detection using the open-source SEN12-FLOOD dataset. By extracting topological features from each image and incorporating them into neural networks, we demonstrate that topological descriptors carry meaningful flood signals independently and complement existing networks to yield more robust and interpretable flood detection systems.

Thin-structure segmentation--power lines, cracks, lane markings at 1-3 pixel width--requires preserving connectivity that standard representations preclude: patching severs continuous structures and conventional superpixels merge thin targets into background before classification. Topology-aware losses penalize connectivity breaks at the objective level but cannot recover what the representation has already destroyed. We propose SEMIR, a framework that replaces the pixel lattice with a parameterized graph minor whose contraction map preserves thin-structure connectivity under the contraction criterion. The minor collapses millions of pixels into tens or hundreds of boundary-aligned supernodes, enabling full-resolution inference without patching at scales demonstrated up to 21 MP in this paper; a lightweight GNN classifies the reduced graph and an exact map lifts predictions to pixel resolution. One pipeline--identical architecture, features, loss, and GNN hyperparameters across all dataset--matches or exceeds domain-specific baselines on TTPLA (power lines), CrackSeg9k (pavement cracks), and SkyScapes Lane (aerial markings) on Dice, IoU, and Boundary F1 while reducing mask fragmentation by at least 4.6x relative to SLIC at matched inference.

While Diffusion Transformers (DiTs) have revolutionized high-fidelity video generation, their reliance on 3D full attention creates a quadratic computational bottleneck. Existing sparse methods face a dilemma: dynamic pruning suffers from prohibitive runtime overhead and memory fragmentation, while static heuristics fail to capture fine-grained dependencies. In this work, we propose ScalingAttention, a training-free framework grounded in a key inductive bias: while individual activations are input-dependent, the high-mass attention regions for each head rapidly converge to a stable, prompt-agnostic Intrinsic Sparse Topology. This topology is weight-encoded, scale-invariant, and efficient to extract. ScalingAttention decouples topology discovery from sparsity control via: (1) WEST (Weight-Encoded Sparse Topology), which extracts a robust block-sparse prior mask offline to eliminate runtime search; (2) FAST (Fidelity-Aware Sensitivity Tuning), which adaptively tunes head-wise sparsity based on diffusion fidelity requirements. To ensure practical acceleration, we co-design a hardware-aligned bit-wise block-sparse kernel. Experiments on Wan2.1 show up to 1.90X end-to-end speedup with superior fidelity, establishing a new Pareto frontier over state-of-the-art baselines.

While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion. Across benchmark datasets including MNIST, QM-3D, and SKEMPI WT, we demonstrate that this reduced feature space captures the essential predictive signal of the full spectrum, and in some cases outperforms it, while significantly reducing computational overhead and preventing the noise introduced by higher-frequency eigenvalues. Our results suggest that these invariants provide a principled, fixed-length interface between spectral geometry and topological learning.

We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines $p$-adic numbers with topological data analysis. Each DNA sequence is encoded along two complementary axes: a $p$-adic distance on $k$-mer prefixes, which captures hierarchical positional structure, and a compositional $L_1$ distance on $k$-mer frequencies, which captures local sequence content. The two distances jointly parameterise a bi-filtered Vietoris--Rips complex, and per-sequence topological summaries from this bi-filtration serve as features for standard machine learning classifiers. We establish theoretical guarantees for the construction: stability under metric perturbations and invariance to the choice of prime, alongside a result that explains why a single $p$-adic axis is topologically uninformative and why the bi-filtration recovers nontrivial homology. On twelve genomic benchmarks ($28$ to $500$ sequences, $3$ to $7$ classes), pVR outperforms four established alignment-free baselines on three of six low-sample datasets, with gains of up to $21$ percentage points; it underperforms only on a SARS-CoV-2 variant benchmark whose point-mutation divergence violates the hierarchical assumption, and all methods saturate in the large-sample regime. pVR also outperforms zero-shot frozen embeddings from the 500M-parameter Nucleotide Transformer v2 by $6.7$ to $11.4$ percentage points on three low-sample benchmarks. The pVR codebase is publicly available at https://github.com/MAHI-Group/pVR.

This study introduces an automated deep learning framework for predicting brain injury (BI) in preterm infants from T2-weighted MRI (dHCP dataset). We propose GloResNet, a lightweight 3D CNN based on ResNet-10, pretrained on MedicalNet to address data scarcity. A global manifold mapping strategy first resamples each 3D volume to 128x128x128 and then applies subject-wise z-score intensity normalization, thereby preserving global topology while standardizing appearance. Training integrates mixup, class weighting, and test-time augmentation for robustness. In 5-fold cross-validation, GloResNet achieved 75.18% average accuracy (peak 81.82%), with specificity 0.81 and sensitivity 0.76. Results demonstrate that a topology-aware lightweight CNN has the capability to effectively predict neonatal BI, offering a non-invasive screening tool. The source code of this paper can be obtained from the GitHub repository: https://github.com/ICL-SUST/GloResNet-Preterm-Brain

Coordination graphs are a central abstraction in cooperative multi-agent reinforcement learning (MARL), yet existing sparse-graph learners lack a theoretically grounded mechanism to decide which edges should exist and how much information each edge should carry. Current methods rely on heuristic criteria that offer no formal guarantee on the learned topology, and no principled way to allocate different communication capacities to structurally different agent relationships. To address this, we propose Heterogeneous Information-Bottleneck Coordination Graphs (HIBCG), which learns a group-aware sparse graph in which both edge existence and message capacity are theoretically justified. With the graph information bottleneck (GIB) serving as the underlying tool, HIBCG first constructs a group-aligned block-diagonal prior that provides a closed-form criterion for edge retention -- determining which edges should exist and at what density per group block -- and then controls per-agent feature bandwidth on the resulting topology, compressing messages to retain only task-relevant content. We prove that the group-aligned prior strictly tightens the variational bound on topology learning, that the objective decomposes per group block, enabling differential edge control, and that capacity allocation follows a water-filling principle.

Hyperdimensional (HD) computing offers an attractive alternative to deep networks for edge learning due to its simplicity, fast prototype-based inference, and compatibility with online updates. However, standard pixel-based HD encoders are brittle: small distribution shifts such as rotation, noise, or occlusion can drastically reduce accuracy. We extract discrete topological primitives-most notably holes-from binarized shapes and pair them with rotation/translation/scale (RTS)-invariant shape signatures. Our method constructs RTS-stable descriptors for (i) the outer shape using a spatial-pyramid variant of Zernike moments and (ii) each hole using an intrinsic Fourier descriptor of its radial signature together with RTS-canonical relative geometry. Each primitive is mapped to a bipolar hypervector via randomized projection and role binding, and variable-cardinality hole sets are aggregated by permutation-invariant bundling to form a single image hypervector. To avoid over-weighting any cue, we learn nonnegative reliability weights for the Zernike and hole channels on a validation set via late fusion of cosine similarities. Experiments on MNIST and EMNIST under controlled corruptions (rotation, Gaussian noise, salt-and-pepper, cutout, zoom) show that Topology-guided HD computing substantially improves robustness compared with a naive HD baseline, maintaining high accuracy across multiple corruption families and benefiting from lightweight online training. Compared with a compact CNN trained on clean data, our method achieves competitive clean accuracy while offering markedly stronger robustness to several pixel-level corruptions, demonstrating that explicit topological structure is a practical route to robust HD representations. The code is provided at https://github.com/arpan-kusari/Topological-HDC.

The Nearest Class Mean (NCM) classifier is widely favored in Class-Incremental Learning (CIL) for its superior resistance to catastrophic forgetting compared to Fully Connected layers. While Neural Collapse (NC) theory supports NCM's optimality by assuming features collapse into single points, non-linear feature drift and insufficient training in CIL often prevent this ideal state. Consequently, classes manifest as complex manifolds rather than collapsed points, rendering the single-point NCM suboptimal. To address this, we propose Hierarchical-Cluster SOINN (HC-SOINN), a novel classifier that captures the topological structure of these manifolds via a ``local-to-global'' representation. Furthermore, we introduce Structure-Topology Alignment via Residuals (STAR) method, which employs a fine-grained pointwise trajectory tracking mechanism to actively deform the learned topology, allowing it to adapt precisely to complex non-linear feature drift. Theoretical analysis and Procrustes distance experiments validate our framework's resilience to manifold deformations. We integrated HC-SOINN into seven state-of-the-art methods by replacing their original classifiers, achieving consistent improvements that highlight the effectiveness and robustness of our approach. Code is available at https://github.com/yhyet/HC_SOINN.

Accurate High-Definition (HD) map construction is critical for autonomous driving, yet existing methods face a fundamental trade-off: vectorization-based approaches preserve topology but struggle with geometric fidelity, while rasterization-based approaches enable precise geometric supervision but produce unstructured outputs. To bridge this gap, we propose GSMap, a novel framework that unifies both paradigms via a learnable 2D Gaussian representation. Each map element is modeled as an ordered sequence of 2D Gaussians, whose centers correspond to the vertices of the vectorized polyline/polygon. This formulation enables simultaneous optimization through: (1) Differentiable rasterization that enforces pixel-level geometric constraints, and (2) Topology-aware vectorization that maintains structural regularity. Experiments on both nuScenes and Argoverse2 demonstrate that our Gaussian-based representation effectively unifies geometric and topological learning, achieving significant performance improvements and demonstrating strong compatibility with existing HD mapping architectures. Code will be available at https://github.com/peakpang/GSMap

After a somewhat rocky start, geometry and topology have established a foothold in machine learning. Message passing, either on graphs or higher-order complexes, is one of the main drivers of geometric deep learning, and paradigms that were once considered to be firmly in the realm of the abstract-like sheaves-have been "tamed" to serve as novel inductive biases for model architectures in topological deep learning. The veritable diversity of models, however, is in stark contrast to the scarcity of suitable benchmark datasets. As a result, researchers often resort to lifting existing graph datasets to include higher-order information. In this opinion paper, I want to encourage the community to also source new datasets, which may be used to prop up the foundations of our research field.

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations, such as structural refinements, that are not appropriate for graph data. In this work, we extend MANTRA, a benchmark dataset containing manifold triangulations, to a larger class of manifolds with more diverse homeomorphism types. We show that, unlike prior claims, both graph neural networks (GNNs) and higher-order message passing (HOMP) methods can saturate the benchmark. However, we find that this is contingent on the right representation and feature assignment, emphasizing their importance in baseline models. We thus provide a novel evaluation protocol based on representational diversity and triangulation refinement. Surprisingly, we find no indication that existing models are capable of generalizing beyond the combinatorial structure of the data. This points towards a research gap in developing models that understand topological structure independent of scale. Our work thus provides the necessary scaffolding to evaluate future models and enable the development of topology-aware inductive biases.

Understanding the topology of decision regions is central to explaining the inner workings of deep neural networks. Prior empirical work has provided evidence that these regions are path connected. We study a stronger topological question: whether closed loops inside a decision region can be contracted without leaving that region. To this end, we propose an iterative quad-mesh filling procedure that constructs a finite-resolution label-preserving surface bounded by a given loop and lying entirely within the same decision region. We further connect this construction to natural Coons patches in order to quantify its deviation from a canonical geometric interpolation of the loop. By evaluating our method across several modern image-classification models, we provide empirical evidence supporting the hypothesis that decision regions in deep neural networks are not only path connected, but also simply connected.

Scientific time series often encode predictive geometric structure, including connectivity, cycles, shell-like geometry, directional changes, and nonlinear neighborhoods, that standard dot-product attention does not explicitly represent. We introduce a topology-aware attention framework that adds such structure to attention logits using persistent homology (H0-H2), anchored Euler characteristic transforms, and kernel-Hilbert channels. A validation-gated local residual captures local topological signals, including a Zeng-style local H0 component, only when held-out validation data support the correction. Exact Vietoris-Rips computations and smooth topological surrogates are evaluated under a no-leakage protocol with train-only calibration, validation-only selection, and test-only reporting. We evaluate guarded topology-aware variants across three architecture families: lightweight attention/Ridge, PatchTSTForRegression, and TimeSeriesTransformerForPrediction. Experiments include synthetic benchmarks isolating higher-order topology and real datasets covering CO2, S&P 500 return-window geometry, and NASA IMS bearing degradation. The audit uses matched paired comparisons across seven dataset units, three random seeds, and three chronological splits, giving 63 paired units per architecture and 189 paired units overall. Topology-aware models show positive paired effects when geometry is predictive, with heterogeneous magnitude across datasets and architectures. Lightweight attention/Ridge improves in 46 of 63 units, with mean relative RMSE reduction of 12.5% and paired randomization p=7.2e-4; PatchTST improves in 33 units and retains the baseline in 20 units, with 23.5% reduction and p=3.5e-5; and TimeSeriesTransformer improves in 47 units, with 47.8% reduction and p<1e-4. The results support topology as a validation-selected, architecture-compatible inductive bias.

This study proposes a novel approach to extract topological properties, specifically the Euler characteristic, from input images using neural networks without relying on large pre-existing datasets but with a single geometric image. Inspired by solid-state physics, where topological properties of magnetic structures are derived from spin field analysis, our model generates a unit vector field from an image, interpreted as a spin configuration. The Euler characteristic is then predicted by computing the skyrmion number of this generated spin configuration. Remarkably, the network learns to construct chiral magnetic textures without access to ground-truth chiral spin configurations, relying instead on only a single, simple geometric image and the straightforward skyrmion number computation. Furthermore, spin configurations generated by independently trained networks can be non-unique due to inherent degrees of freedom. To constrain these degrees of freedom and further refine the spin configuration, we incorporate a magnetic Hamiltonian, comprising exchange interaction, Dzyaloshinskii-Moriya (DM) interaction, and anisotropy, as an additional, physics-informed loss function. We validate the model's efficacy on complex geometrical shapes and demonstrate its applicability to practical tasks.

Topological features play an essential role in ensuring geometric plausibility and structural consistency in image analysis tasks such as segmentation and skeletonization. However, integrating topology-preserving learning based on simple points into deep learning tasks remains challenging, as existing simple point detection methods are confined to binary images and are non-differentiable, rendering them incompatible with gradient-based optimization in modern deep learning. Moreover, morphological and purely data-driven approaches often fail to guaranty topological consistency. To address these limitations, we propose a novel method that directly computes simple points on continuous-valued images, enabling differentiable topological inference. Building on this theory, we develop an efficient skeleton extraction algorithm that preserves topological structures in binary and continuous-valued images. Furthermore, we design a variational model that enforces topological constraints by preserving topologically non-removable (i.e., non-simple) points, which can be seamlessly integrated into any deep neural network segmentation with softmax or sigmoid outputs. Experimental results demonstrate that the proposed approach effectively improves topological integrity and structural accuracy across multiple benchmarks. The codes are available in https://github.com/levnsio/CSP.

Accurate multiclass segmentation of the Circle of Willis (CoW) is essential for neurovascular disease management but remains challenging due to complex vascular topology and variable morphology. Existing deep learning methods often suffer from vascular discontinuities and inter-class misclassification, while current topological loss functions incur prohibitive computational costs in 3D multiclass settings. To address these limitations, we propose an Anatomically-Guided Topology-Aware Loss (AG-TAL) and introduce a large-scale, multi-center CoW dataset with unified annotations to facilitate robust model training. AG-TAL specifically integrates a radius-aware Dice loss to address class imbalance in small vessels, a breakage-aware clDice loss that utilizes group convolutions to efficiently preserve local connectivity, and an adjacency-aware co-occurrence loss that leverages anatomical priors to enforce distinct boundaries between neighboring arteries. Evaluated using 5-fold cross-validation, AG-TAL achieved an average Dice score of 80.85% for all CoW arteries, with small arteries notably higher by 1.05-3.09% compared to state-of-the-art methods. Across six independent datasets, the performance of AG-TAL achieved Dice scores ranging from 74.46% to 81.17% for all CoW arteries, with improvements of 2.20% to 9.98% for small arteries compared to other methods. This study demonstrates the superiority of AG-TAL in identifying multiclass CoW arteries and its ability to generalize well to multiple independent datasets. Furthermore, reliability analyses and clinical applications in an Alzheimer's disease cohort validate the AG-TAL's robustness and its potential for discovering imaging-based morphological biomarkers.

Reliable celestial attitude determination is a critical requirement for autonomous spacecraft navigation, yet traditional "Lost-in-Space" (LIS) algorithms often suffer from high computational overhead and sensitivity to sensor-induced noise. While deep learning has emerged as a promising alternative, standard regression models are often confounded by the non-Euclidean topology of the celestial sphere and by the periodic boundary conditions of Right Ascension (RA) and Declination (Dec). In this paper, we present Star-Fusion, a multi-modal architecture that reformulates orientation estimation as a discrete topological classification task. Our approach leverages spherical K-Means clustering to partition the celestial sphere into K topologically consistent regions, effectively mitigating coordinate wrapping artifacts. The proposed architecture employs a tripartite fusion strategy: a SwinV2-Tiny transformer backbone for photometric feature extraction, a convolutional heatmap branch for spatial grounding, and a coordinate-based MLP for geometric anchoring. Experimental evaluations on a synthetic Hipparcos-derived dataset demonstrate that Star-Fusion achieves a Top-1 accuracy of 93.4% and a Top-3 accuracy of 97.8%. Furthermore, the model exhibits high computational efficiency, maintaining an inference latency of 18.4 ms on resource-constrained COTS hardware, making it a viable candidate for real-time onboard deployment in next-generation satellite constellations.

Topology reasoning is crucial for autonomous driving. Current methods primarily focus on instance-level learning for centerline detection, followed by a sequential module for topology reasoning that relies on simplified MLP layers. Moreover, they often neglect the importance of \textit{point-to-instance} (P2I) relationships in topology reasoning. To address these limitations, we present TopoHR (Topological Hierarchical Representation), a novel end-to-end framework that establishes cyclic interaction between centerline detection and topology reasoning, allowing them to iteratively enhance each other. Specifically, we introduce a hierarchical centerline representation including point queries, instance queries, and semantic representations. These multi-level features are seamlessly integrated and fused within a hierarchical centerline decoder. Furthermore, we design a hierarchical topology reasoning module that captures both fine-grained P2I relationships and global instance-to-instance (I2I) connections within a unified architecture. With these novel components, TopoHR ensures accurate and robust topology reasoning. On the OpenLane-V2 benchmark, TopoHR refreshes state-of-the-art performance with significant improvements. Notably, compared with previous best results, TopoHR achieves +3.8 in $\mathrm{DET}_{\text{l}}$, +5.4 in $\mathrm{TOP}_{\text{ll}}$ on \text{subset_A} and +11.0 in $\mathrm{DET}_{\text{l}}$, +7.9 in $\mathrm{TOP}_{\text{ll}}$ on \text{subset_B}, validating the effectiveness of the proposed components. The code will be shared publicly at https://github.com/Yifeng-Bai/TopoHR.git.

We generalise the reparameterization trick (RT) applied in variational autoencoders (VAEs) letting these have latent spaces of non-trivial topology - i.e. that of base manifolds covered with other ones, on which some technique for RT is available. That is possible since covering maps are measurable - moreover, this allows to establish an inequality on KL-divergence between pushforward (PF) densities on the base latent manifold, bounding it with KL-divergence between pullbacks on the cover, in some cases making the KL-term of VAE's ELBO analytically tractable, despite the topological non-triviality of the supporting latent manifold. Our development follows a route close but somewhat alternative to reparameterization on Lie groups, the latest proposal for which is to reparameterize PFs of normal densities from the Lie algebra - "through" the exponential map, seen by us as a particular case of what we propose to call reparameterization via covering (RVC). We demonstrate the working of our approach by constructing a VAE with the latent space of Klein bottle (not a Lie group) topology, which we call KleinVAE, successfully learning an appropriate artificial dataset. We discuss potential applicability of such topology-informed generative models as weight priors in Bayesian learning, particularly for convolutional vision models, where said manifold was peculiarly shown to have some relevance.

Text-attributed graphs integrate semantic information of node texts with topological structure, offering significant value in various applications such as document classification and information extraction. Existing approaches typically encode textual content using language models (LMs), followed by graph neural networks (GNNs) to process structural information. However, during the LM-based text encoding phase, most methods not only perform semantic interaction solely at the word-token granularity, but also neglect the structural dependencies among texts from different nodes. In this work, we propose DuConTE, a dual-granularity text encoder with topology-constrained attention. The model employs a cascaded architecture of two pretrained LMs, encoding semantics first at the word-token granularity and then at the node granularity. During the self-attention computation in each LM, we dynamically adjust the attention mask matrix based on node connectivity, guiding the model to learn semantic correlations informed by the graph structure. Furthermore, when composing node representations from word-token embeddings, we separately evaluate the importance of tokens under the center-node context and the neighborhood context, enabling the capture of more contextually relevant semantic information. Extensive experiments on multiple benchmark datasets demonstrate that DuConTE achieves state-of-the-art performance on the majority of them.

In artificial-intelligence-aided signal processing, existing deep learning models often exhibit a black-box structure. Here, conceptually beyond spectral analysis, we demonstrate that topological methods not only effectively capture intrinsic and complex structural information but can also enhance neural networks. We provide a transparent methodology, TopCap, to capture topological features inherent in time series for basic machine learning. Compared to prior approaches, we obtain descriptors that probe finer information such as the vibration of a time series. Notably, in classifying voiced and voiceless consonants, TopCap achieves an accuracy consistently standing in comparison with neural network models. Moreover, by integrating TopCap features into those neural networks, our approach improves upon state-of-the-art methods in terms of robustness against noise, as well as accuracy, stability, convergence of loss function, and interpretability.