Topological Data Analysis

Flowcharts are widely used in industrial requirements, but usually remain embedded as static images. Vision Language Models (VLMs) show promise in the conversion of these flowcharts into machine-readable models for RE activities, yet, when directly applied to flowchart conversion, they often fail on topology-critical visual details. To address this, we propose EdgeFlow that augments a VLM's original input with a deterministically extracted Canny edge map-acting as a structural prior-to improve flowchart-to-Mermaid conversion, without requiring annotated training data or domain-specific model fine-tuning. We evaluate EdgeFlow on IndusReqFlow, a dataset sourced from real-world requirements. Compared with off-the-shelf VLMs, EdgeFlow improves node-level F1 by 17.39 percentage points and edge-level F1 by 16.94 percentage points. At the path level, EdgeFlow improves path F1 by 11.06 percentage points, enabling better support for model-based testing. These results demonstrate that EdgeFlow provides a practical, training-free means to improve topology-preserving flowchart-to-Mermaid conversion for industrial RE. Cross-dataset evaluation results on a public synthetic benchmark show no significant improvement; this highlights the need for diverse benchmarks incorporating industrial data for the comprehensive evaluation of future VLM-based RE tools.

Advancements in 3D curve skeletonization are accelerating progress across a wide range of applications. However, developing robust skeletonization algorithms that capture intricate object details remains challenging. Skeletonization via Local Separators (LS) offers an efficient graph-based approach but suffers from representation inaccuracies due to its discrete nature. To address this, we introduce CSCD, a novel framework for Curve Skeletonization in the Continuous Domain, generalizing LS to manifolds. Specifically, we present two realizations: CSCD-M for meshes and CSCD-PC for point clouds. CSCD-M leverages the intrinsic triangulation of a mesh for resilience to noise and improved topological preservation, while CSCD-PC employs tufted Laplacians for enhanced robustness. To our knowledge, CSCD-M is the first intrinsic method for curve skeletonization. Our results show CSCD-M matches LS performance across diverse meshes and outperforms LS (TOG'21) on benchmarks like Thingi10k dataset. CSCD-PC qualitatively outperforms CoverageAxis++ (Eurographics'24) and EPCS (CAG'23). Finally, we demonstrate the efficacy of CSCD in a few downstream tasks: object classification, shape segmentation, identifying handles, tunnels, and constrictions in objects. Project Website: https://cscd-skel.pages.dev

Neural networks encode inputs as high-dimensional vectors, known as representations, that capture how models process data by encoding task-relevant structure and semantics. Representation alignment refers to the degree to which different models, layers, or training conditions produce similar representations for the same inputs, with important implications for model interpretation, selection, and robustness analysis. Existing approaches to measure alignment primarily rely on geometric properties, such as neighborhood and cluster similarity, offering limited insight into the global organization of representations. In this work, we present TopoAlign, a topology-aware framework for visually comparing model representations from a structural perspective. Leveraging mapper graphs from topological data analysis, TopoAlign jointly analyzes graphs constructed from representations of shared inputs across different models or layers. The framework supports a top-down comparative workflow: it first performs global structure alignment via joint force-directed optimization to produce coordinated graph layouts; it then identifies local correspondences through automated detection of structurally matching regions, visualized with Bubble Sets; and finally it enables fine-grained pattern inspection through motif-based queries and membrane-inspired visualizations. We demonstrate TopoAlign through case studies on language and multimodal models, complemented by expert feedback. Our results show that TopoAlign provides meaningful insights into representation structure and alignment from a topological perspective.

Vascular circulation follows fundamental biophysical principles that optimize mass transport and metabolic energy expenditure, which can be effectively modeled by Murray's law. However, contemporary deep learning methods for vascular segmentation often neglect these biophysical constraints. This leads to physiologically implausible branching and misclassification vascular trees, rendering. These automated segmentation results are unreliable unreliable for downstream clinical tasks such as blood flow simulation or disease quantification. In this paper, we introduce MARVEL (Universal MurrAy's law-infoRmed Vessel sEgmentation and topoLogy estimation), a backbone-agnostic framework that integrates biophysical priors into vascular tree extraction. MARVEL combines per-pixel supervision with explicit radius predictions to enforce local bifurcation constraints derived from an empirical width-exponent mapping. We implement these constraints as differentiable regularizers during training to guide models toward physiologically consistent reconstructions. We evaluate MARVEL on eight public datasets across multiple vascular modalities and segmentation backbones. Results demonstrate MARVEL's superior performance in segmentation accuracy, topological consistency, and physiological plausibility. By converting segmented masks into graph-based hemodynamic simulations, we demonstrate that MARVEL preserves the subtle pathological narrowing and topological connectivity required to distinguish hypertensive from normotensive eyes. Results show that MARVEL significantly improves the classification of hypertension via arteriovenous pressure differences in the eye (p < 0.001), outperforming baseline models in both topological consistency and clinical predictive value.

Robotic manipulation of deformable linear objects (DLOs) presents significant challenges due to complex dynamics and frequent self-occlusions. Existing robotic knot tying methods typically rely on precise topological state tracking with ordered keypoints and explicit edge connectivity. This reliance makes them prone to failures due to tracking drift and topology mismatch caused by repeated bending and crossings during knot formation.To address these limitations, we introduce RoboHitch, a novel framework that learns to perform hitch knot tying from human demonstrations using only disordered 3D keypoints and RGB images. This eliminates the need for explicit topological order, allowing for more flexible manipulation. Our method employs a dynamic Graph Autoencoder to extract geometric features from untracked keypoints, complemented by a Convolutional Autoencoder that captures essential visual context. A bidirectional cross-attention mechanism then fuses these modalities to jointly predict pick and place affordances, facilitating implicit reasoning about the rope's state and enabling knot tying under occlusion.Real-world experiments demonstrate the effectiveness and generalizability of our approach, successfully completing hitch knots in scenarios with self-occlusions.

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

Phylogenetic trees are hybrid objects: branch lengths vary continuously, while topologies change discretely through edge contractions and expansions. Billera-Holmes-Vogtmann (BHV) tree space provides a canonical geometry for this structure, representing each resolved topology as a Euclidean orthant and topological changes as motion across shared lower-dimensional boundaries. We introduce PhylaFlow, a hybrid flow-matching model that learns posterior-basin transport in BHV tree space. PhylaFlow is trained on BHV geodesic paths from random starting trees to short-run posterior samples, coupling continuous branch-length motion within orthants with learned boundary events and discrete topology transitions. We evaluate the learned geometry operationally: if the flow reaches posterior-relevant regions, finite-budget Bayesian refinement initialized from, or guided by, its terminal trees should recover posterior-supported topologies more efficiently. Across DS1-DS8 phylogenetic posterior benchmarks, PhylaFlow substantially reduces initial Tree-KL relative to classical initializers. After finite-budget MrBayes refinement, direct PhylaFlow improves early and intermediate topology-recovery trajectories on most datasets, while split-guided PhylaFlow-MCMC obtains the strongest hard-case results. The best PhylaFlow variant outperforms short-warmup on seven of eight datasets and PhyloGFN on five of eight under the same refinement budget. In a joint sequence-conditioned experiment, sequence embeddings steer posterior split recovery, although exact posterior topology recovery remains preliminary. These results show that hybrid flow matching can learn actionable transport in BHV tree space and provide a geometry-aware proposal mechanism for Bayesian phylogenetic inference.

The exponential growth of global academic output has confronted researchers and AI agents with an unprecedented information explosion,'' where fragmented and unstructured knowledge organization impedes deep interdisciplinary integration. Current academic retrieval tools predominantly rely on superficial keyword matching or vector-space semantic retrieval, which lack the topological reasoning capabilities required to navigate complex logical connections. Agentic deep-research-based frameworks are often prone to logical hallucinations and consuming high inference costs. To bridge this gap, in this report, we introduce SciAtlas, a large-scale, multi-disciplinary, heterogeneous academic resource knowledge graph designed as a panoramic scientific evolution network. By integrating over 43M papers from 26 disciplines, and a total of 157M entities and 3B triplets, SciAtlas provides a structured topological cognitive substrate that dismantles disciplinary barriers and furnishes AI agents with a global perspective. Furthermore, we develop a neuro-symbolic retrieval algorithm featuring tri-path collaborative recall and graph reranking, achieving a seamless transition from simple semantic matching to deterministic association discovery. We also present key application directions of SciAtlas, including literature review, automated research trend synthesis, idea positioning, and academic trajectory exploration, to demonstrate that SciAtlas can serve as an effective cognitive map'' to empower the full loop of automated scientific research while significantly reducing reasoning costs. We have released the interfaces for KG retrieval and various downstream tasks in our GitHub repo.

AI agents deployed into SRE workflows currently derive their understanding of environment state from raw observability telemetry at query time, paying a semantic-interpretation tax in tokens, latency, and inferential reliability. We propose Causely, a causal intelligence layer that maintains a structured representation of environment topology, attribute dependencies, and causal relationships that are anchroed to a ontological representation of the managed environment. Causely transforms raw telemetry into a live, queryable model providing the semantic and causal foundation AI agents require to diagnose, evaluate impact, and act safely in production. We evaluate this value proposition through a benchmark study conducted in a controlled setting with injected faults in a 24-microservice OpenTelemetry demo application. Our experiments compare four agent configurations (Claude Code, OpenAI Codex, HolmesGPT with Sonnet and Gemini backends). Experiments are run with and without access to Causely under two scenarios: an active incident and a healthy baseline. On the active-fault scenario, causal grounding reduces mean time-to-diagnosis by 63%, mean token consumption by 60%, and mean tool-call count by 78%, compressing the investigation footprint by 4.8$\times$ and lowering direct API cost per run by 57%; root-cause-diagnosis accuracy rises from 75% to 100%.

We introduce \textbf{Prism}, a framework for structural symmetry diagnosis in complex networks. Given a graph Laplacian $L$ and a duality operator $P$ (a symmetric involution), Prism computes the \emph{duality defect} $δ(L,P) = \|LP - PL\|_F / \|L\|_F$ -- a scalar measuring how far the network deviates from structural self-consistency. When $P$ encodes the network's true symmetry, $δ$ starts near zero and rises monotonically as structure degrades; an arbitrary $P$ gives noise. We prove that the optimal $L'$ satisfying $[L', P] = 0$ is given by a closed-form block-diagonal projection, and provide an unsupervised alternating optimization that learns $P$ from the graph's own Fiedler vector. Experiments on synthetic networks show the true-$P$ defect is $3.38\times$ more sensitive to structural degradation than an index-reversal baseline and more sensitive than modularity. On Zachary's Karate Club with edge noise, Prism achieves $94.5\%$ community detection accuracy at $5\%$ noise versus $76.6\%$ for the raw Laplacian baseline. Applied to live S&P~500 data (2026-05-17), Prism detects rising structural stress (defect $0.43 \to 0.73$ over 90 days) while surface correlations remain low -- a signal invisible to correlation-based methods. In a historical backtest spanning five major stress events (2011--2020), the duality defect exhibits a consistent pattern: it reaches elevated levels \emph{before} the correlation spike that accompanies each crisis, and sustains high readings during periods of structural fragility that conventional metrics classify as calm. The duality defect is a first-principles structural admissibility condition, requiring no training data and computable in milliseconds.

Flow in porous media is difficult to address using standard analytical or numerical methods due to its complexity. However, since synthetic representations of porous media are easy to produce and data from physical experiments are becoming more widely available, the problem is well-suited to studies that include machine learning (ML) techniques. We discuss a number of features that can be extracted from such data, and their utility as input variables into a standard ML algorithm. These features include structural measures describing the geometry of the porous media, topological measures describing the connectivity, and network measures obtained by modeling the porous media as simplified pore networks. These features enable the prediction of the permeability of the considered (synthetic) porous materials using ML techniques that also leverage the separately computed exact permeability (ground truth). Comparing results obtained using different input variables helps develop a better understanding of the utility of various measures for predicting permeability based on the porous media structure. We show, in particular, that topological data analysis (TDA) provides a useful set of features that can be easily combined with ML to yield meaningful results.

Dimensionality reduction algorithms map high-dimensional data into visualizable 2D or 3D spaces, but traditionally rely on a discrete point-cloud paradigm. This discrete abstraction is susceptible to visual occlusion and artificial discontinuities, often failing to represent the continuous density of the underlying manifold. To address these limitations, we introduce Topo-GS, a framework that repurposes 3D Gaussian Splatting (3DGS) to cast multidimensional projection as a meshless volumetric reconstruction process. Instead of standard photometric losses, Topo-GS is driven by local geometric constraints. By solving orthogonal Procrustes targets, the optimization enforces an As-Rigid-As-Possible prior while explicitly aligning the spatial covariance of each Gaussian to the local tangent space. Recognizing that unrolling data of varying intrinsic dimensionalities requires distinct spatial treatments, we utilize a topology-aware strategy that tailors the loss formulation to preserve either continuous 1D trajectories or cohesive 2D surfaces. Quantitative and visual evaluations demonstrate that Topo-GS successfully transforms discrete scatter plots into continuous volumetric representations, where inherent projection distortions explicitly manifest as observable geometric variations, while preserving local topological fidelity comparable to discrete baselines.

Recent advances in Diffusion Transformers (DiTs) demonstrate that aligning noisy latent states with well-trained semantic features-as pioneered by Representation Alignment (REPA)-can substantially accelerate training and improve generation fidelity. Subsequent analysis(e.g., iREPA) suggests that these gains arise primarily from transferring spatial structure contained in pre-trained vision representations. However, mostly existing alignment methods employ point-wise matching objectives or rely on implicit architectural tweaks, which fail to explicitly model the spatial relational geometry inherent in vision foundation models. We argue that such element-wise supervision is insufficient to capture the rich spatial topology of visual representations, and that effective alignment for generation should instead be formulated as an explicit structural constraint. To this end, we propose sREPA, a structural REPresentation Alignment framework to enforce consistency in the relational geometry of feature maps, rather than merely matching individual feature points. By encouraging the model to internalize holistic spatial layouts and structural correlations from pre-trained features, sREPA achieves faster and more stable convergence, along with improved sample quality, compared to state-of-the-art alignment strategies. Our code and models will be released.

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.

Large vision-language models have significantly advanced GUI agents, enabling executable interaction across web, mobile, and desktop interfaces. Yet these gains largely rely on a forgiving region-tolerant paradigm, where many nearby pixels inside the same component remain valid. Precise geometric construction breaks this assumption: actions must land on points in continuous canvas space rather than tolerant regions. Because geometric primitives carry ontological dependencies, a local coordinate error can induce cascading topological failures that distort downstream objects and invalidate the final construction. We identify this regime as precision-sensitive GUI tasks, requiring point-level accuracy, geometry-aware verification, and robustness to dependency-driven error propagation. To benchmark it, we introduce PAGE Bench, with 4,906 problems and over 224K process-supervised, pixel-level GUI actions. We further propose PAGER, a topology-aware agent that decomposes construction into dependency-structured planning and pixel-level execution. Pixel-grounded supervised tuning establishes executable action grammar, while precision-aligned reinforcement learning mitigates rollout-induced exposure bias through state-conditioned geometric feedback. Experiments reveal a pronounced Semantic-Execution Gap: general multimodal models can exceed 88% action type accuracy yet remain below 6% task success. PAGER closes this gap, delivering 4.1x higher task success than the strongest evaluated general baseline and raising step success rate from below 9% for GUI-specialized agents to over 62%, establishing a new state of the art for point-precise GUI control.

This paper develops a feedback-based method to preserve the topology privacy of consensus protocols in network systems. The key idea is to intentionally violate topology identifiability conditions, thereby preventing unique or accurate recovery of the true topology from available observations, while preserving the intended consensus behavior. This problem is challenging because the feedback magnitude directly reflects the privacy level of edges, while it is strongly coupled with the consensus convergence and constrained by local communications at each node. To begin with, we derive the feedback conditions of both partial and full observation cases, where the topology unsolvability from observation data is characterized in the former, and the solution space that enforces topology inaccuracy from data is constructed in the latter. Then, we propose a novel distributed topology modification design under limited privacy budgets, and establish the performance guarantees through a controllable tradeoff between the consensus deviation and the topology privacy. Finally, we develop a low-complexity heuristic algorithm to achieve optimal privacy preservation on existing edges. Comparative simulations validate the effectiveness and outperformance of the proposed preservation design.

Root cause analysis (RCA) in microservices is challenging due to (i) noisy and heterogeneous multimodal observability (metrics, logs, traces), (ii) cascading failure propagation that amplifies downstream symptoms, and (iii) non-stationary topology drift induced by autoscaling and rolling updates. Recent LLM-based RCA agents can generate tool-grounded explanations, yet they often remain topology-agnostic and suffer from \emph{symptom-amplification bias}, misattributing the root cause to salient downstream victims. We propose \textbf{TopoEvo}, a topology-aware self-evolving multi-agent framework that couples graph representation learning with structured, topology-constrained reasoning. TopoEvo first introduces \emph{Metric-orthogonal Multimodal Alignment} (MOMA), which decomposes metric embeddings into complementary subspaces and contrastively aligns logs and traces to reduce modality redundancy and sparsity, yielding stable node representations for graph encoding. It then applies \emph{Vector Quantization} (VQ) to discretize topology-enhanced states into auditable \emph{symptom tokens} with a symptom lexicon, enabling reliable retrieval and token-level evidence grounding. On top of these discrete topology cues, TopoEvo performs a multi-agent \emph{Hypothesis--Evidence--Test} (HET) workflow to explicitly verify propagation-consistent explanations and separate initiating anomalies from amplified downstream symptoms. Finally, a \emph{Self-Evolving Mechanism} refreshes hierarchical incident memory and performs conservative test-time adaptation with high-confidence pseudo-labels to maintain robustness under drift.

Half a billion citation edges extracted from 100.7 million Ukrainian court decisions reveal that judicial citation structure encodes legal domain boundaries without supervision and predicts future legislative importance with near-perfect accuracy. We construct the first large-scale citation graph from the complete EDRSR registry (99.5 million full texts, 1.1 TB), extracting 502 million citation links across six types via regex on commodity hardware in approximately 5 hours, with precision of 1.00 on a 200-decision validation sample (95% Wilson CI: [0.982, 1.000]). Three principal findings emerge. (1) The degree distribution follows a power law (alpha = 1.57 +/- 0.008), placing the Ukrainian court network near the EU Court of Justice and below the US Supreme Court, with hub articles cited by millions of decisions. (2) Louvain community detection on the co-citation projection recovers legal domain boundaries (civil, criminal, administrative, commercial) with modularity Q = 0.44-0.55 and temporal stability (NMI = 0.83-0.86 across periods), constituting an automatically constructed legal ontology grounded in judicial practice. (3) Citation features predict top-1000 articles with AUC = 0.9984, substantially outperforming a naive frequency baseline (P@1000 = 0.655); temporal dynamics detect legislative regime changes as phase transitions and the 2022 invasion as a citation entropy spike (H: 11.02 -> 13.49) with emergent wartime legislation nodes. The citation-derived ontology is operationalized as the domain layer of a workflow memory system for LLM-assisted legal analysis, connecting to the ontology-controlled paradigm. The extraction pipeline, analysis code, and aggregated statistics are released as open data.

We introduce TopoPrimer, a framework that makes the global topological structure of the series population an explicit input to any forecasting model. TopoPrimer improves accuracy across diverse domains, stabilizes forecasts under seasonal demand spikes, and closes the cold-start gap. Precomputed once per domain via persistent homology and spectral sheaf coordinates, TopoPrimer deploys per token for fully-trained models and as a lightweight adapter for pre-trained backbones. Of these two components, sheaf coordinates are the primary accuracy driver. Across four public benchmarks on Chronos and TimesFM, TopoPrimer consistently improves forecasting accuracy, with gains of up to 7.3% MSE on ECL. The topology advantage persists with near-identical magnitude across zero-shot and fine-tuned backbones, suggesting topology and per-series training capture complementary signals. The gains are most pronounced in difficult regimes. Under peak seasonal demand, classical and zero-shot models degrade by up to 50%, while TopoPrimer stays within 10%. At cold start with no item history, TopoPrimer reduces MAE by 27% over a topology-free baseline.

High-fidelity 3D head generation plays a crucial role in the film, animation and video game industries. In industrial pipelines, studios typically enforce a fixed reference topology across all head assets, as such a clean and uniform topology is a prerequisite for production-level rigging, skinning and animation. In this paper, we present TOPOS, a framework tailored for single image conditioned 3D head generation that jointly recovers geometry and appearance under such an industry-standard topology. In contrast to general 3D generative models which produce triangle meshes with inconsistent topology and numerous vertices, hindering semantic correspondence and asset-level reuse, TOPOS generates head meshes with a fixed, studio-style topology, enabling consistent vertex-level correspondence across all generated heads. To model heads under this unified topology, we proposed a novel variational autoencoder structure, termed TOPOS-VAE. Inspired by multi-model large language models (MLLMs), our TOPOS-VAE leverages the Perceiver Resampler to convert input pointclouds sampled from head meshes of diverse topologies into the target reference topology. Building upon TOPOS-VAE's structured latent space, we train a rectified flow transformer, TOPOS-DiT, to efficiently generate high-fidelity head meshes from a single image. We further present TOPOS-Texture, an end-to-end module that produces relightable UV texture maps from the same portrait image via fine-tuning a multimodal image generative model. The generated textures are spatially aligned with the underlying mesh geometry and faithfully preserve high-frequency appearance details. Extensive experiments demonstrate that TOPOS achieves state-of-the-art performance on 3D head generation, surpassing both classical face reconstruction methods and general 3D object generative models, highlighting its effectiveness for digital human creation.

While generative models have shown promise in pediatric sleep analysis, the latent structure of their multimodal embeddings remains poorly understood. This work investigates session-wide diagnostic information contained in the sequences of 30-second pediatric PSG epochs embedded by a multimodal masked autoencoder. We test whether augmenting embeddings with PHATE-derived per-epoch coordinates and whole-night movement descriptors, persistent homology summaries of the embedding cloud, and EHR yields task-relevant signals. Simple linear and MLP models, chosen for interpretability rather than state-of-the-art performance, show that geometric, topological, and clinical features each provide complementary gains. For binary predictions, feature importance is task-dependent, and more expressive late-fusion models generally perform better, with AUPRC improving from 0.26 to 0.34 for desaturation, 0.31 to 0.48 for EEG arousal, 0.09 to 0.22 for hypopnea, and 0.05 to 0.14 for apnea. We also report Brier score and Expected Calibration Error, where the full fusion model yields the best calibration across all four binary tasks. Our study reveals that latent geometry/topology and EHR offer complementary, interpretable signals beyond embeddings, improving calibration and robustness under extreme imbalance.

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of \textbf{756 images} of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, Gemini 3.1 Pro, achieves only \textbf{71.1%} tree-generation accuracy on CurveBench-Easy and \textbf{19.1%} on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained Qwen3-VL-8B model improves over \texttt{Qwen-3-VL-8B-Thinking} from \textbf{2.8%} to \textbf{33.3%} tree-generation accuracy on CurveBench-Easy, exceeding GPT-5.4 and Claude Opus 4.5 under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.

Category-agnostic pose estimation (CAPE) aims to localize keypoints on query images from arbitrary categories, using only a few annotated support examples for guidance. Recent approaches either treat keypoints as isolated entities or rely on manually defined skeleton priors, which are costly to annotate and inherently inflexible across diverse categories. Such oversimplification limits the model's capacity to capture instance-wise structural cues critical for accurate pixel-level localization. To overcome these limitations, we propose GenCape, a Generative-based framework for CAPE that infers keypoint relationships solely from image-based support inputs, without additional textual descriptions or predefined skeletons. Our framework consists of two principal components: an iterative Structure-aware Variational Autoencoder (i-SVAE) and a Compositional Graph Transfer (CGT) module. The former infers soft, instance-specific adjacency matrices from support features through variational inference, embedded layer-wise into the Graph Transformer Decoder for progressive structural priors refinement. The latter adaptively aggregates multiple latent graphs into a query-aware structure via Bayesian fusion and attention-based reweighting, enhancing resilience to visual uncertainty and support-induced bias. This structure-aware design facilitates effective message propagation among keypoints and promotes semantic alignment across object categories with diverse keypoint topologies. Experimental results on the MP-100 dataset show that our method achieves substantial gains over graph-support baselines under both 1- and 5-shot settings, while maintaining competitive performance against text-support counterparts.

Replicating and surpassing the autonomy of natural organisms remains a long-standing goal in robotics. Yet most robotic systems have their structure, materials, and control designed separately, in sharp contrast to the co-evolution in nature. This separation often leads to suboptimal designs, and we still have a limited understanding of the individual and collective contributions of these design entities. In this work, we propose a gradient-based co-design framework that simultaneously optimizes the topology, material distribution, and control policy of a truss-lattice robot. The framework embeds mixed-type topological and material variables into a continuous design space and integrates a neural network controller within a differentiable simulator, capturing their interactions and enabling efficient gradient calculation via automatic differentiation. Furthermore, we develop a constrained optimization to navigate the highly non-convex design landscape and jointly optimize all design entities. Case studies demonstrate that the proposed framework consistently discovers diverse locomotion strategies that outperform baselines obtained through separated design. The framework is also flexible to accommodate different functional requirements and boundary conditions. Using this framework, we further extract design insights that reveal the individual and collective effects of different entities on robotic performance. The proposed framework provides a computational foundation for the autonomous co-design of robotic systems, capable of reconfiguration, locomotion, and other complex autonomous behaviors.

Differentiable vector graphics have enabled powerful gradient-based optimization of vector primitives directly from raster images. However, existing frameworks formulate this as a flat optimization problem, forcing hundreds to thousands of randomly initialized curves to blindly compete for pixel-level error reduction. This disordered optimization leads to topology collapse, where macroscopic structures are distorted by internal high-frequency noise, resulting in a redundant and uneditable "polygon soup" that limits practical editability. To address this limitation, we propose Vector Scaffolding, a novel hierarchical optimization framework that shifts from flat pixel-matching to structured topological construction tailored for vector graphics. By identifying a key cause of topology collapse as the mathematical imbalance between area and boundary gradients, we introduce Interior Gradient Aggregation to stabilize the learning dynamics of multi-scale curve mixtures. Upon this stabilized landscape, we employ Progressive Stratification and Rapid Inflation Scheduling to progressively densify vector primitives with extremely high learning rates ($\times 50$). Experiments demonstrate that our approach accelerates optimization by $2.5\times$ while simultaneously improving PSNR by up to 1.4 dB over the previous state of the art.

Graph Neural Networks (GNNs) have achieved strong results in molecular property prediction, but polymers present distinct challenges: labeled datasets are scarce and small (typically in the order of hundreds of polymers) due to the need for expensive experimentation, and complex polymer chain distributions influence polymer properties. Established practice in polymer prediction represents polymers solely by graphs of their repeat units, discarding the chain-scale morphology that governs key properties such as the glass transition temperature ($T_g$). In this work, we propose a principled graph construction that addresses this gap. Given a polymer's molecular mass distribution (MMD), we sample representative chains from the Schulz-Zimm distribution and construct representative sets of large graphs encoding chain-scale topology directly, with atoms and bonds featurized using rich chemical descriptors. We further pretrain GNN encoders via masked graph modeling on 100,000 unlabeled PSMILES strings before fine-tuning on labeled data. On a dataset of 381 polymers (180 homopolymers and 201 copolymers), we show that graph construction and self-supervised pretraining are jointly necessary: without pretraining, the large graph method matches the repeat-unit baseline (28.40 K vs. 28.36 K RMSE); with pretraining, it achieves 24.76 K +/- 3.30 K, a 5.1% reduction in mean error over the pretrained repeat-unit baseline (26.08 K +/- 4.20 K, p < 0.001, 30 runs). An ablation removing chemical features degrades performance to 36.65 K, confirming both components are essential. Results are architecture-agnostic, holding for both GINE and GATv2 encoders.

Multi-agent systems (MAS) have emerged as a promising paradigm for solving complex tasks. Recent work has explored self-evolving MAS that automatically optimize agent capabilities or communication topologies. However, existing methods either learn a topology that remains fixed at inference time or adapt only the topology or capability during inference. We empirically and theoretically show that effective test-time evolution requires jointly adapting both axes, but on different time scales: capabilities should update rapidly to handle emerging subtasks, while the topology should evolve more slowly to preserve coordination stability. We then introduce TacoMAS, a test-time co-evolution framework for dynamic MAS. TacoMAS formulates MAS inference as a task of online graph adaptation, where nodes represent agents with role-specific capabilities and edges define their communication topology. During inference, a fast capability loop updates agent expertise using trajectory-level feedback, while a slow meta-LLM-driven topology loop performs agents' birth-death operations on MAS, including edge edit, agent addition, and agent removal. We further show that this fast-slow design drives MAS evolution toward a task-conditioned stable equilibrium. Experiments on four benchmarks demonstrate that TacoMAS outperforms nearly 20 multi-agent baselines, achieving an average improvement of 13.3% over the strongest baseline. The codes are released at https://github.com/chenxu2-gif/TacoMAS-MultiAgent.

We present FrameTwin, a curve-anchored Gaussian alignment framework that uses sparse-view images to close the control loop for adaptive wireframe 3D printing. Our key idea is to capture the deformation of thin wireframe structures from sparse-view images using Gaussian kernels anchored to parametric curves, yielding a compact and geometry-aware encoding that explicitly captures strut topology. Driven by a differentiable rendering pipeline, FrameTwin estimates a neural deformation field that aligns the partially printed target model with the deformed structure observed during fabrication, where the optimized curve-Gaussian representation serves as a digital twin of the evolving wireframe. Unlike general Gaussian-splatting approaches, our formulation constrains kernel placement along parametric curves, substantially reducing the ambiguity inherent in sparse-view observations of thin structures. The resultant deformation-field alignment enforces global consistency across all struts. By using the estimated deformation field to blend the distorted printed geometry with the remaining unprinted geometry, FrameTwin enables adaptive updates to future printing trajectories. We demonstrate that FrameTwin can robustly capture and compensate for deformation in wireframe models fabricated using a robotized 3D printing system.

Imitation learning is effective for training agents when expert demonstrations are available, but collecting demonstrations for every complex task in an environment is costly. We study the long-horizon, goal-conditioned setting where a fixed demonstration dataset contains useful behavior, but not complete examples for every task the agent must solve. Existing imitation learning methods can learn strong policies from demonstrations, but when solving long-horizon tasks, small errors accumulate over long primitive-action trajectories and make zero-shot adaptation to new tasks unreliable. We introduce Zero-shot Agents from Latent Topologies (ZALT), an imitation-learning method that solves unseen start-goal tasks beyond those demonstrated during training. ZALT identifies latent hub states where trajectories converge or diverge, learns policies and a dynamics model over hub-to-hub transitions, and plans over the hub topology to complete new tasks. This topology makes demonstrated behaviors explicitly composable while compressing long tasks into shorter sequences of abstract transitions -- combined, these enable ZALT to perform zero-shot adaptation. In a complex 3D maze environment, ZALT achieves 55% zero-shot success on unseen tasks, compared to 6% for the strongest baseline.

Graph neural networks (GNNs) are increasingly used to model biological systems, yet the reliability of post-hoc explanation methods for recovering meaningful molecular mechanisms remains unclear. Here, we systematically evaluate four widely used approaches: Saliency Attribution (SA), Integrated Gradients (IG), GNNExplainer, and Layer-wise Relevance Propagation (LRP) for identifying disease-relevant structure in breast cancer RNA-seq data projected onto a protein-protein interaction network. Using synthetic benchmarks with known ground-truth motifs, we show that explanation methods recover distinct signal organizations: SA performs best for sparse single-node drivers, whereas IG and LRP preferentially recover distributed pathway-like and cascade-like signals. In TCGA BRCA data, we identify a consistent topological signature of disease-associated hubs in which attribution peaks in the immediate 1-hop neighborhood and decays across successive network shells, a pattern most pronounced for IG and LRP and associated with strong enrichment of known cancer hubs. We further observe a trade-off between local hub enrichment and global gene ranking performance, with IG optimizing local enrichment and SA achieving superior global discrimination. Motivated by these complementary behaviors, we introduce a framework combining a shell-based hub score with consensus ranking across explainers. Consensus scores improve prioritization of canonical cancer genes (TP53, BRCA1, ESR1, MYC), reduce dependence on node degree, and, especially when tuned, outperform individual methods. Pathway enrichment further reveals improved recovery of biologically coherent cancer programs, including ERBB2, RTK, MAPK, immune, and cytokine signaling. Together, these results demonstrate that topology-aware integration of graph explanations can improve biological interpretability and biologically relevant molecular recovery.

Classic exploration methods often rely on dense occupancy maps or high-resolution point clouds for frontier detection and path planning, resulting in substantial memory consumption and computational overhead. Moreover, micro UAVs under size, weight, and power (SWaP) constraints are not practical to be equipped with sensors like LiDAR to obtain accurate environmental geometric measurements. This paper presents a lightweight autonomous exploration system that leverages omnidirectional vision and sparse topological map guidance. Specifically, we utilize a multi-fisheye camera setup to achieve omnidirectional Field of View (FoV) and perform depth estimation. To address the limited depth estimation accuracy, frontiers are represented as potential unexplored regions characterized by topological nodes instead of explicit boundaries, enabling efficient identification of frontier regions without maintaining occupancy grids or global point clouds. Unlike classic dense representations, our approach abstracts the environment using a sparse topological map composed of key nodes and their descriptors, reducing memory consumption and computational demands. Global path planning is performed directly on the sparse graph. The proposed method is validated in both simulation and on a palm-sized vision-based UAV with an 11 cm wheelbase and a 400 g weight in real-world experiments, demonstrating that our method can achieve efficient exploration with extremely low computational consumption.

Decades of cognitive science establish that humans navigate environments by forming cognitive maps, defined as allocentric and topology-preserving representations of 3D space. While modern Vision-Language Models (VLMs) demonstrate emergent spatial reasoning from 2D egocentric inputs, it remains unclear whether they construct an analogous 3D internal representation. In this paper, we demonstrate that current VLMs do possess a latent topological map of 3D scenes, but it is heavily overshadowed by non-geometric visual semantics, such as color and shape. By isolating this spatial subspace through cross-scene linear feature extraction, we extract a clean spatial subspace that causally controls the model's spatial outputs. We mathematically shape this latent representation and prove its correspondence to the Laplacian eigenmaps of the scene's 3D Gaussian-kernel graph, converging to the physical 3D space in the continuous limit. Motivated by this geometric identification, we further introduce a mathematically principled latent regularization method for VLMs, based on Dirichlet energy. Applying this single-term regularizer to a minimal 500-step supervised VLM fine-tuning (SFT) on simple synthetic data yields significant improvements on real-world spatial benchmarks, outperforming standard SFT and competitive baselines by up to 12.1% in spatial tasks involving scene topology understanding. Source code is available at https://github.com/pittisl/vlm-latent-shaping

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.

We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology ($H_1$). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

Multi-agent LLM systems for code generation face a fundamental routing problem: the optimal orchestration topology depends on the structural complexity of the code under modification, yet existing systems select topologies without consulting the codebase. We present Retrieval-Guided Adaptive Orchestration (RGAO), an architecture that closes this loop by extracting a structural complexity vector from a hierarchical code index before selecting the orchestration topology. RGAO operates within Code-Agent, a multi-agent framework whose sub-agents are governed by formal contracts with six-dimensional budget vectors. Our headline contribution is the composition of two previously separate lines of work -- complexity-conditioned LLM routing and formal resource algebras -- yielding a property neither admits alone: provable budget conservation under retrieval-conditioned dynamic topology selection. Concretely we contribute: (1) a complexity-conditioned topology router that reduces proxy-measured misrouting from 30.1% to 8.2%; (2) a budget algebra with a structural-induction conservation theorem; and (3) a hierarchical code retrieval engine. Empirical evaluation demonstrates sub-millisecond DAG construction and linear tree-index scalability.

We present a single-image head mesh reconstruction framework that addresses the longstanding challenge of simultaneously preserving facial identity and producing industry-grade topology. Our framework adopts a coarse-to-fine optimization pipeline that refines a rigged template across three stages -- rig, joint, and vertex -- achieving stable convergence and consistent topology. To mitigate the ill-posed nature of single-image 3D face reconstruction and ensure identity preservation, we employ a normal consistency objective jointly with landmark alignment. To further preserve local surface structure and enforce topological regularity, we introduce geometry-aware constraints based on Gaussian curvature and conformal consistency, along with auxiliary regularizations that correct fine artifacts such as lip seams and eyelid discontinuities. Our hierarchical optimization with geometry-aware regularization yields meshes with semantically meaningful edge flow and industry-grade topology. After geometry reconstruction, we extract UV-space texture and normal maps to preserve appearance details for visualization and downstream use. In a user study with 22 professional technical artists, our results were assessed as approaching industry-grade usability, and 95% of participants ranked our method as the top-performing approach, underscoring its effectiveness for real-world digital human production.

We propose a topology-constrained quantized nnUNet framework for efficient and anatomically accurate 3D tooth segmentation, addressing the challenges of spatial distortion introduced by quantization in deep learning models. The proposed method integrates a novel tooth-specific topological loss into quantization-aware training, preserving critical anatomical structures such as tooth count, adjacency relationships, and cavity integrity while maintaining computational efficiency. The system employs an 8-bit quantized nnUNet backbone, where weights and activations are dynamically calibrated to minimize precision loss during inference. Furthermore, the topological loss combines connected-component analysis, adjacency consistency, and hole detection penalties, ensuring anatomical fidelity without modifying the underlying network architecture. The joint optimization objective harmonizes cross-entropy loss, quantization regularization, and topological constraints, enabling end-to-end training with gradient approximations for persistent homology terms. Experiments demonstrate that our approach significantly reduces topological errors compared to conventional quantized models, achieving clinically plausible segmentations on dental CBCT scans. The method retains the hardware efficiency of integer-only inference, making it suitable for deployment in resource-constrained clinical environments. This work bridges the gap between computational efficiency and anatomical precision in medical image segmentation, offering a practical solution for real-world dental applications.

We study day-ahead transmission topology control for high-voltage grid operation under $N-1$ security constraints. The operational task is to select, over a 24-hour horizon, a sequence of substation topologies obtained via busbar-coupler switching to relieve line overloads while limiting switching effort and topological complexity. We formulate this task as a sequential multi-objective optimization problem with four objectives used in TSO decision making: worst-case $N-1$ line loading, maximum topological depth, number of topology changes, and time spent outside the reference topology. We propose an exact block algorithm that exploits the temporal structure of topology plans: consecutive hours with the same topology are represented as blocks, enabling enumeration of the complete Pareto front over the admissible set of topologies under fixed operational bounds on depth and switching. We also develop a tailored NSGA-III-based evolutionary heuristic and evaluate it against the exact front. Using real operational data from the Dutch high-voltage transmission grid operated by TenneT, the block algorithm computes the exact front for a highly congested day in under three minutes after topology-level load-flow preprocessing. The exact front reveals low-switching plans with no DC $N-1$ thermal overloads that the tested evolutionary search fails to find. The proposed method, therefore, provides both a practical day-ahead decision-support tool for transmission operators and a benchmark for heuristic and learning-based topology-control methods.

Continual learning systems face a fundamental tension between plasticity -- acquiring new knowledge -- and stability -- retaining prior knowledge. We introduce MPCS (Multi-Plasticity Continual System), a neuroplastic architecture that integrates eleven complementary mechanisms: task-driven neurogenesis, Fourier-encoded inputs, EWC regularization, meta-replay, mixed consolidation, hybrid gating, synapse pruning/regeneration, Hebbian updates, task similarity routing, adaptive growth control, and continuous neuron importance tracking. We evaluate MPCS on MEP-BENCH, a multi-track benchmark spanning 31 tasks across regression, classification, logic, and mixed domains, using a three-dimensional Pareto criterion over task performance (Perf), representation diversity (RD), and gradient conflict rate (GCR). Across 15 ablation configurations (3 seeds x 4 tracks x 2000 epochs), MPCS achieves a Normalized Efficiency Score of 94.2, placing it on the Pareto frontier among 9 of 14 gate-passing systems. Key findings: (i) Fourier encoding is the single most critical component (removal drops Perf by 30.7 pp and fails the MEP gate on 14% of tasks); (ii) global EWC degrades performance (NES = -4.2); topology-local EWC reduces this penalty (NES 90.5->91.8) but does not eliminate it; removing EWC entirely yields MPCS_EFFICIENT, the highest-Perf system -- establishing a monotone relationship in the high task-similarity regime (s_bar ~= 0.95): global EWC < topology EWC < no EWC; (iii) the Pareto status assessment is predictive: removing the two Pareto-dominated components (EWC + Hebbian) jointly yields MPCS_EFFICIENT, which improves Perf by 0.6 pp at 4.7x lower compute cost (127 vs. 602 min), validating the Pareto frontier as an actionable model-compression guide.

Automated medical report generation, MRG, holds substantial value for alleviating radiologist workload and enhancing diagnostic efficiency. However, mainstream approaches typically treat diverse chest abnormalities as isolated classification targets. This paradigm often overlooks inherent disease co-occurrences and struggles to translate medical topological structures into explicit data correlations, constraining the model's reasoning capacity on complex or subtle lesions. To address this, we propose a Graph-Augmented Dual-Stream Medical Report Generation with Topological Internalization, GDMRG. Our framework introduces a Topological Knowledge Internalization module, TKI, which leverages a Graph Convolutional Network, GCN, to generate an explicit parameterized weight matrix based on global disease co-occurrence priors. This facilitates efficient topological knowledge injection without relying on external retrieval mechanisms. Building upon this, we construct a dual-stream classification system: the main branch generates discrete diagnostic prompts under topological constraints, while the auxiliary branch employs an asymmetric optimization strategy to dynamically calibrate decision boundaries for highly imbalanced samples. Concurrently, to establish a logical closed loop between diagnosis and visual grounding, we design a diagnostic-driven Diagnosis-Guided Spatial Attention, DGSA, that utilizes high-dimensional clinical semantics to recalibrate the visual encoder, mitigating feature hallucinations. Comprehensive experiments on the MIMIC-CXR dataset demonstrate that GDMRG achieves competitive clinical efficacy, CE, while maintaining natural language fluency. Furthermore, our model exhibits robust zero-shot generalization on the IU X-Ray dataset. In summary, this work presents an integrated and interpretable paradigm for medical report generation.

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.

Blood vessel segmentation and -tracing are essential tasks in many medical imaging applications. Although numerous methods exist, the prevailing segment-then-fix paradigm is fundamentally limited regarding its suitability for modeling the task of complete and topologically accurate vascular network reconstruction. Here, we propose an approach to extract topologically more accurate vascular graphs from 3D image data, building upon highly successful ideas from the related biomedical tasks of cell segmentation and -tracking. Our approach first predicts voxel-wise vessel direction vectors joint with standard vessel segmentation masks. Second, to extract the vascular graph from these predictions, we introduce a direction-vector-guided extension of the TEASAR algorithm. Our approach achieves state-of-the-art performance on three benchmark datasets, spanning both synthetic and real imagery. We further demonstrate the applicability of our approach to challenging 3D micro-CT scans of rat heart vasculature. Finally, we propose meaningful and interpretable measures of topological error, namely false splits and false merges for graphs. Overall, our approach substantially improves the topological accuracy of reconstructed vascular graphs, being able to separate closely apposed vessel segments and handle multiple vascular trees within a single volume.

Mixture-of-experts (MoE) architectures have turned LLM serving into a cluster-scale workload in which communication consumes a considerable portion of LLM serving runtime. This has prompted industry to invest heavily in expensive high-bandwidth scale-up networks. We question whether such costly infrastructure is strictly necessary. We present the first systematic cross-layer analysis of network cost-effectiveness for MoE LLM serving, comparing four representative XPU (e.g., GPU/TPU) topologies (scale-up, scale-out, 3D torus, and 3D full-mesh). We find that lower-cost switchless topologies are more cost-effective than the scale-up topology across all serving scenarios explored, improving cost-effectiveness by 20.6-56.2%. In particular, the 3D full-mesh topology is Pareto-optimal in terms of the performance-cost tradeoff. We also find that current scale-up link bandwidths are over-provisioned: reducing the link bandwidth improves throughput per cost by up to 27%. A forward-looking analysis of upcoming GPU generations indicates that the cost-performance advantage of switchless networks will likely persist.

Aligning large language models (LLMs) with human preferences is commonly done via reinforcement learning from human feedback (RLHF) with Proximal Policy Optimization (PPO) or, more simply, via Direct Preference Optimization (DPO). While DPO is stable and RL-free, it treats preferences as flat winner vs. loser signals and is sensitive to noisy or brittle preferences arising from fragile chains of thought. We propose TUR-DPO, a topology- and uncertainty-aware variant of DPO that rewards how answers are derived, not only what they say, by eliciting lightweight reasoning topologies and combining semantic faithfulness, utility, and topology quality into a calibrated uncertainty signal. A small learnable reward is factorized over these signals and incorporated into an uncertainty-weighted DPO objective that remains RL-free and relies only on a fixed or moving reference policy. Empirically, across open 7-8B models and benchmarks spanning mathematical reasoning, factual question answering, summarization, and helpful/harmless dialogue, TUR-DPO improves judge win-rates, faithfulness, and calibration relative to DPO while preserving training simplicity and avoiding online rollouts. We further observe consistent gains in multimodal and long-context settings, and show that TUR-DPO matches or exceeds PPO on reasoning-centric tasks while maintaining operational simplicity.

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.

As one of the mainstream models of artificial intelligence, world models allow agents to learn the representation of the environment for efficient prediction and planning. However, classical world models based on flat tensors face several key problems, including noise sensitivity, error accumulation and weak reasoning. To address these limitations, many recent studies use graph structure to decompose the environment into entity nodes and interactive edges, and model virtual environments in a structured space. This paper systematically formalizes and unifies these emerging graph-based works under the concept of graph world models (GWMs). To the best of our knowledge, GWMs have not yet been explicitly defined and surveyed as a unified research paradigm. Furthermore, we propose a taxonomy based on relational inductive biases (RIB), categorizing GWMs by the specific structural priors they inject: (1) spatial RIB for topological abstraction; (2) physical RIB for dynamic simulation; and (3) logical RIB for causal and semantic reasoning. For each model category, we outline the key design principles, summarize representative models, and conduct comparative analyses. We further discuss open challenges and future directions, including dynamic graph adaptation, probabilistic relational dynamics, multi-granularity inductive biases, and the need for dedicated benchmarks and evaluation metrics for GWMs.

Heterogeneous graphs are widely used to model multi-relational systems, but missing node attributes remain a major bottleneck for downstream learning. In this paper, we identify and formalize type-dependent information asymmetry: the phenomenon that different node types provide substantially different levels of useful signal for attribute completion. Motivated by this observation, we propose TypeBandit, a lightweight, model-agnostic methodology for heterogeneous attribute completion. TypeBandit combines topology-aware initialization, type-level bandit sampling, and joint representation learning. It allocates a finite global sampling budget across node types, samples representative nodes within each type, and uses the resulting sampled type summaries as shared contextual signals during representation construction. By operating at the type level rather than over each target node's local neighborhood, TypeBandit keeps the adaptive state compact and practical for large heterogeneous graphs. A key advantage of TypeBandit is architectural flexibility. Rather than requiring a new heterogeneous graph neural network architecture, TypeBandit acts as a type-aware front end for representative heterogeneous GNN backbones, including R-GCN, HetGNN, HGT, and SimpleHGN. We further introduce a hybrid pretraining scheme that combines structural degree priors with feature propagation, yielding a more reliable initializer than degree-only pretraining. Under a fixed-split protocol on DBLP, IMDB, and ACM, TypeBandit provides dataset-dependent but practically meaningful gains. Additional ablation, stability, efficiency, semantic-propagation, and sampled OGBN-MAG experiments support TypeBandit as a practical strategy for heterogeneous attribute completion when type-specific information is unevenly distributed and sampling resources are limited.

Representational collapse, where embeddings become anisotropic and lose multi-scale structure, can erode downstream performance long before performance metrics react. We propose an online, topology-aware monitor for evolving neural representations that couples Modular Morse Homology Maintenance (MMHM) with a composite Collapse Index (CI). Instead of rebuilding complexes each epoch, we apply sparse edits at a fixed scale and maintain a discrete Morse matching, yielding fast, incremental updates. Across LLM fine-tuning and temporal KGE training, CI provides a low-latency early-warning signal suitable for in-training interventions. Code and experimental scripts will be released publicly

Continuous causal discovery typically couples representation learning with structural optimization via non-convex acyclicity penalties, which subjects solvers to local optima and restricts scalability in high-dimensional regimes. We propose a decoupled paradigm that shifts the causal discovery bottleneck from non-convex optimization to statistical score estimation. We introduce the Score-Schur Topological Sort (SSTS), an algorithm that extracts topological order directly from unconstrained generative models, bypassing constrained structure optimization. We establish that the causal hierarchy leaves a geometric signature within the score function: iterative graph marginalization is mathematically equivalent to computing the Schur complement of the Score-Jacobian Information Matrix (SJIM) under linear conditions. This translates the acyclicity constraint into an algebraic procedure with a dominant cost of O(d^3) operations. For non-linear systems, we formulate the expectation gap of Schur marginalization and introduce Block-SSTS to compress extraction depth, bounding structural error. Empirically, SSTS allows causal structural analysis on non-linear graphs up to d=1000. At this scale, our framework indicates that once the non-convex optimization bottleneck is mathematically bypassed, the structural fidelity of continuous causal discovery is bounded by the finite-sample estimation variance of the global score geometry. By reducing graph extraction to matrix operations, this work reframes scalable causal discovery from a constrained optimization problem to a statistical estimation challenge.

Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We show that standard local metrics reward this noise memorisation: top-performing embeddings invent cycles and disconnected islands absent from the data. We introduce a topology-faithfulness benchmark based on noisy manifolds with known homology, tune DiRe against it, and find Pareto-optimal configurations that match or beat GPU-accelerated UMAP on classification while recovering exact first Betti numbers on stress tests. On 723K arXiv paper embeddings, DiRe preserves 3-4 times more topological structure than UMAP at comparable wall-clock.

The continuous advancement of autonomous driving (AD) introduces challenges across multiple disciplines to ensure safe and efficient driving. One such challenge is the generation of High-Definition (HD) maps, which must remain up to date and highly accurate for downstream automotive tasks. One promising approach is the use of crowdsourced data from a vehicle fleet, representing road topology and lane-level features. This work focuses on the generation of centerlines and lane dividers from crowdsourced vehicle trajectories. We adopt a Detection Transformer (DETR)-based approach, where a rasterized representation of vehicle trajectories is used as input to predict vectorized lane representations. Each lane consists of a centerline with an associated direction and corresponding lane dividers that are geometrically constrained by the centerline. Our method includes the extraction of local tiles, from which crowdsourced vehicle trajectories are aggregated. Each tile undergoes a transformation into a rasterized representation encoding both the presence and direction of each trajectory, enabling the prediction of vectorized directed lanes. Experiments are conducted on an internal dataset as well as on the public datasets nuScenes and nuPlan.

Graph neural ordinary differential equations (Graph ODEs) extend graph learning from discrete message-passing layers to continuous-time representation flows. While it supports adaptive long-range propagation, we show that Graph ODEs with strictly positive irreducible mixing operators face an inherent \emph{monostability trap}: in the long-time regime, information leakage is unavoidable and the dynamics converge to a single global consensus attractor. We propose the \textbf{Hysteresis Graph ODE (HGODE)}, which couples feature evolution with a latent topological potential driven by a learned pairwise force. A double-well edge potential and bipolarized gate allow edge states to polarize into connected or insulated phases while preserving differentiability. We provide asymptotic analysis of the collapse mechanism and the proposed hysteretic topology dynamics, and validate HGODE on theory-driven synthetic diagnostics and real-world graph benchmarks.

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.

A levelset free but a hybrid image segmentation approach based on a modified version of the piece wise constant shape gradient of an Mumford Shah shape functional and a repulsive function is considered. The segmentation is performed a non-local shape based through an evolution of discrete curves driven by a non local shape based energy to segment images containing disjoint regions and multiple boundaries. This formulation has a novel additional component as a multivariable function dependent on a few sampled points of the curves that handles the occurrence of self intersection during boundary curves evolution. The method is applied to a few gray scale and color images, including images with nested structures and astronomical objects. The results indicate effective segmentation in complex scenarios with absolute control on the topology of the segments and self-intersections of the boundaries

This paper studies the computational difficulty of clustering problems that are defined directly on a continuous probability density. Rather than working with finite samples, we assume the density is given as a polynomial and ask whether it contains certain cluster structures. Four natural questions are examined. First, do there exist several points with high density that are far apart from each other. Second, do two high density points have a midpoint with low density, creating a valley between them. Third, does the region where the density is above a threshold have at least a given number of separate connected pieces. Fourth, does that same region contain a hole, meaning a loop that cannot be shrunk to a point. We prove that the first two problems, separated points and valley detection, are exactly as hard as the existential theory of the reals, a complexity class that contains NP and is believed to be strictly larger. In contrast, the topological problems of counting connected pieces and detecting holes are at least as hard as the existential theory of the reals, but their exact complexity remains open. Placing them inside that class would need a major advance in real algebraic geometry. These results give the first rigorous classification of exact continuous clustering inside the real polynomial hierarchy. They also show that even basic clustering criteria are not NP complete unless unexpected collapses occur.

Recent advances in pretraining 3D point cloud encoders (e.g., Point-BERT, Point-MAE) have produced powerful models, whose abilities are typically evaluated on geometric or semantic tasks. At the same time, topological descriptors have been shown to provide informative summaries of a shape's multiscale structure. In this paper we pose the question whether topological information can be derived from features produced by 3D encoders. To address this question, we first introduce DONUT, a synthetic benchmark with controlled topological complexity, and propose FILTR (Filtration Transformer), a learnable framework to predict persistence diagrams directly from frozen encoders. FILTR adapts a transformer decoder to treat diagram generation as a set prediction task. Our analysis on DONUT reveals that existing encoders retain only limited global topological signals, yet FILTR successfully leverages information produced by these encoders to approximate persistence diagrams. Our approach enables, for the first time, data-driven extraction of persistence diagrams from raw point clouds through an efficient learnable feed-forward mechanism.

GBM is a highly aggressive primary malignancy in adults, necessitating personalized therapeutic strategies due to its inherent molecular heterogeneity. MGMT promoter methylation is a pivotal prognostic biomarker for anticipating response to temozolomide-based chemotherapy. Although various AI frameworks have been developed for non-invasive MGMT prediction, spatial heterogeneity of methylation status and the high-dimensional and correlated nature of MRI data frequently constrain discriminative feature learning and generalizability of classical models. To circumvent these limitations, a specialized IA-QCNN architecture is proposed, based on the principles of quantum mechanics, including superposition and entanglement, and enabling more efficient representation learning in high-dimensional Hilbert space. The framework establishes a methodological bridge between GBM radiogenomics and quantum deep learning by integrating energy-based slice selection, importance-aware weighting, ring-topology quantum convolution, and folding-based pooling layers. When the model predicts MGMT promoter methylation status using both mpMRI and T1Gd images, experimental results demonstrate that the IA-QCNN achieves high accuracy despite its low number of trainable parameters while effectively minimizing the overfitting problem observed in classical models. Quantitative analyses reveal that the T1Gd modality possesses higher discriminative power than mpMRI, establishing a clinically significant sequence preference. Furthermore, the model exhibits exceptional robustness in hybrid noise environments, effectively utilizing noise as a regularization mechanism to enhance predictive performance. Consequently, the specialized IA-QCNN architecture provides a robust and computationally efficient alternative to classical approaches in the analysis of heterogeneous radiogenomic data.

Persistent homology, a method from topological data analysis, extracts robust, multi-scale features from data. It produces stable representations of time series by applying varying thresholds to their values (a process known as a \textit{filtration}). We develop novel filtrations for time series and introduce topological methods for the analysis of eye-tracking data, by interpreting fixation sequences as time series, and constructing ``hybrid models'' that combine topological features with traditional statistical features. We empirically evaluate our method by applying it to the task of dyslexia detection from eye-tracking-while-reading data using the Copenhagen Corpus, which contains scanpaths from dyslexic and non-dyslexic L1 and L2 readers. Our hybrid models outperform existing approaches that rely solely on traditional features, showing that persistent homology captures complementary information encoded in fixation sequences. The strength of these topological features is further underscored by their achieving performance comparable to established baseline methods. Importantly, our proposed filtrations outperform existing ones.

Test-time adaptation (TTA) has emerged as a promising paradigm for vision-language models (VLMs) to bridge the distribution gap between pre-training and test data. Recent works have focused on backpropagation-free TTA methods that rely on cache-based designs, but these introduce two key limitations. First, inference latency increases as the cache grows with the number of classes, leading to inefficiencies in large-scale settings. Second, suboptimal performance occurs when the cache contains insufficient or incorrect samples. In this paper, we present Prototype-Based Test-Time Adaptation (PTA), an efficient and effective TTA paradigm that uses a set of class-specific knowledge prototypes to accumulate knowledge from test samples. Particularly, knowledge prototypes are adaptively weighted based on the zero-shot class confidence of each test sample, incorporating the sample's visual features into the corresponding class-specific prototype. It is worth highlighting that the knowledge from past test samples is integrated and utilized solely in the prototypes, eliminating the overhead of cache population and retrieval that hinders the efficiency of existing TTA methods. This endows PTA with extremely high efficiency while achieving state-of-the-art performance on 15 image recognition benchmarks and 4 robust point cloud analysis benchmarks. For example, PTA improves CLIP's accuracy from 65.64% to 69.38% on 10 cross-domain benchmarks, while retaining 92% of CLIP's inference speed on large-scale ImageNet-1K. In contrast, the cache-based TDA achieves a lower accuracy of 67.97% and operates at only 50% of CLIP's inference speed.

Retrieval-Augmented Generation (RAG) has been proposed to mitigate hallucinations in large language models (LLMs), where generated outputs may be factually incorrect. However, existing RAG approaches predominantly rely on vector similarity for retrieval, which is prone to semantic noise and fails to ensure that generated responses fully satisfy the complex conditions specified by factual queries, often leading to incorrect answers. To address this challenge, we introduce a novel research problem, named Exact Retrieval Problem (ERP). To the best of our knowledge, this is the first problem formulation that explicitly incorporates structural information into RAG for factual questions to satisfy all query conditions. For this novel problem, we propose Structure Guided Retrieval-Augmented Generation (SG-RAG), which models the retrieval process as an embedding-based subgraph matching task, and uses the retrieved topological structures to guide the LLM to generate answers that meet all specified query conditions. To facilitate evaluation of ERP, we construct and publicly release Exact Retrieval Question Answering (ERQA), a large-scale dataset comprising 120000 fact-oriented QA pairs, each involving complex conditions, spanning 20 diverse domains. The experimental results demonstrate that SG-RAG significantly outperforms strong baselines on ERQA, delivering absolute improvements from 20.68 to 50.88 points across all evaluation metrics, while maintaining reasonable computational overhead.