Topology

3D grounding aims to localize target entities in complex scenes from natural language and plays a fundamental role in embodied perception and spatial reasoning. However, existing approaches mostly rely on feature similarity or direct matching, making it difficult to connect natural-language intent with the implicit semantic and geometric structures hidden in billion-scale urban point clouds. We reformulate city-scale 3D grounding as structured constraint reasoning, where description semantics are organized into computable cross-modal constraints over open-vocabulary 3D entities, attributes, and spatial relations. We present CitySTAR, a training-free framework for reasoning-driven urban 3D grounding. CitySTAR lifts raw billion-scale urban point clouds into a query-ready scene graph of open-vocabulary 3D instances, with CodeLLM-driven tools supplying multimodal evidence for node attributes and 3D spatial relations. It then models target-context topology with paired hypergraphs and performs bidirectional topology verification for structural disambiguation. Finally, a Reflective Cross-modal Grounding module integrates topology consistency and candidate-centered 2D visual evidence to make decisions over a metric-aware 3D context graph. To further support this setting, we introduce CitySTAR-3D, an enhanced benchmark that improves semantic coverage, instance completeness, bounding-box fidelity, and spatial-relation complexity in city-scale 3D grounding. Extensive experiments show that CitySTAR consistently improves open-world urban 3D grounding while maintaining strong interpretability and generalization.

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.

Active imitation learning reduces expert effort by allowing a learner to request the demonstrations it needs. Existing methods typically select these requests for their expected information gain about the expert policy. In structured multi-task domains, however, the number of start-goal tasks may grow combinatorially despite their solutions sharing reusable behavior. This makes composable behaviors especially valuable, since a single demonstration may help solve many tasks at once. Prior methods do not explicitly account for this value when selecting which demonstration to request. We introduce Adaptive Agents via Latent Topologies (AALT), which requests demonstrations that maximize expected gains in start-goal connectivity. We further show that this objective is formally tied to information gain about task reachability. AALT organizes existing demonstrations into a topology of latent hub states connected by learned behaviors, identifies high-value bridge demonstrations that are likely to enable many tasks at once, and grounds each to an expert query. At inference, it plans through the resulting topology and conditions a diffusion policy on each successive hub transition. In a simulated UR5e robot ordered-retrieval domain with 72 tasks, AALT improved from 42/72 to 72/72 (100%) successful tasks consistently using only 3 demonstrations totaling 5 transitions beyond the initial dataset. After 20 demonstrations, the strongest baseline averaged 88.6% success using 98 transitions.

Message-passing Graph Neural Networks (GNNs) iteratively propagate and aggregate local neighborhood information followed by global readout to learn graph representations. However, their discriminative power is upper-bounded by the Weisfeiler--Lehman (1-WL) graph isomorphism test. This prevents GNNs from distinguishing certain non-isomorphic graphs with identical local neighborhood structures, often leading to similar graph representations. Unified Topological Signatures (UTS) capture compact, multi-scale representation of global graph topology derived from persistent homology. We introduce two complementary UTS signatures: Graph_UTS- a static signature of the input graph topology, and Embedding_UTS- a dynamic signature of the evolving embedding topology. They encode structural information inaccessible to 1-WL-based message-passing GNNs, yet their capabilities are explored solely for post-hoc embedding-space analysis. We integrate UTS into GNN training across three architectural interventions: (i) UTS-Aug: augmenting with standard readout feature that encodes graph's true topology; (ii) UTS-Reg: topological regularizer that constrains representation collapse; (iii) UTS-Pool: topology-guided pooling that retains structurally critical nodes. We further leverage UTS as a layer-wise diagnostic to quantify oversmoothing during GNN training. Theoretically, we show that integrating UTS into GNN optimization strictly extends GNN expressivity beyond the 1-WL hierarchy. Experiments on three graph classification benchmarks show consistent benefits: Graph-UTS, Dual-UTS, and UTS-Pool improve accuracy across all three datasets, Embedding-UTS provides smaller but similarly consistent gains, and UTS-Reg's benefit varies across graph domains. Accuracy improves by up to 5.8% with Graph-UTS augmentation, by up to 1.9% with UTS-Reg, and achieves comparable performance to TOGL with UTS-Pool.

We develop a certified continuation framework for inference and training in deep equilibrium networks (DEQs), with training posed as interpolation to accuracy $2^{-b}$. For inference, input homotopy selects an equilibrium branch from a supplied start root, and a rounded tracker follows it under quantitative conditioning, derivative, boundary, and tube-radius certificates. The framework includes structured factorized certificates, sequential block elimination, inheritance of contraction guarantees in adapted coordinates, and bordered continuation through simple folds. For smooth multidimensional DEQs, including tanh networks, rational local tests can construct and validate oriented continuation charts under explicit geometric promises. For training, programmable dormant bilinear rank-one channels provide output-preserving residual-aligned repairs. Loaded Tikhonov solves diagnose insufficient parameter-to-output directions, while certified gate realization, column stability, well-posed inference, and finite-update error budgets control each pass. Under polynomially bounded certificate, encoding, precision, and backend costs, both inference and training have bit complexity $O(\mathrm{poly}(L+b))$, where $L$ is the encoded instance length; training uses $O(b+\ell)$ passes and reserve channels from an initial residual bounded by $2^\ell$. A budgeted implementation returns either certified success or inconclusive termination. The quantitative core and local certificate machinery are machine-checked in Lean 4, while numerical experiments illustrate the training mechanism.

The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian kernel distances for many pairs of points can be expensive. Using Random Fourier Features (RFF), Chen and Phillips [ALT 2017] showed that for points in a $d$-dimensional Euclidean ball in ${\mathbb R}^N$, $t=Ω((d/\varepsilon^2)\log(dR/\varepsilon))$ features suffice to preserve all pairwise Gaussian kernel distances within a $(1\pm\varepsilon)$ factor with high probability. We establish a uniform relative-error embedding theorem for the more general setting of an arbitrary positive-reach submanifold $\mathcal M\subset{\mathbb R}^N$ of intrinsic dimension $d$. We show that $t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}δ)))$, or approximately $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilonδ))))$, RFFs suffice, with probability $1-δ$, to preserve the Gaussian kernel distance between every pair of manifold points up to relative error $\varepsilon$. Thus the bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate. We also prove a topological consequence: under the same RFF embedding, persistent homology is preserved in the sense that weighted Cech and Rips filtrations built from Gaussian kernel power distance are $(1\pm\varepsilon_\star)$-interleaved, where $\varepsilon_\star$ accounts for both distance distortion and kernel-weight approximation.

Deep learning on cortical surfaces faces a dilemma: capturing the complex topology of over 60 nomenclature-dependent sulci per hemisphere requires high-capacity models, yet the extreme scarcity of expert annotations ($N=62$ subjects) inevitably causes overfitting. Standard supervised approaches fail to generalize in this data-scarce regime, particularly for variable and small sulci where topological ambiguity is high. To overcome this limitation, we introduce a Geometric-to-Semantic Spherical Transfer Learning framework. First, we leverage massive unlabeled data (UK Biobank, $\approx$30,000 subjects) to pre-train a spherical encoder using a locally-optimized strategy. By relying solely on continuous surface features (curvature and depth), the relevance of this pre-training is confirmed by the model's ability to detect localized and rare topological traits, such as sulcal interruptions. The downstream labeling task, however, introduces extracted sulcal fundi (lines) as an explicit semantic input. To bridge this dimensional domain gap (from purely geometric to semantic) without causing catastrophic forgetting, these anatomical lines are integrated into the pre-trained backbone via a soft-initialized Topological Prior Injector. Our experiments demonstrate that this approach outperforms fully supervised baselines trained from scratch, achieving a mean Dice of 0.77. Crucially, a local analysis reveals that the self-supervised geometric priors yield the largest performance gains on variable and tertiary sulci (up to 14.8%), confirming that learning the cortex shape is highly beneficial for identifying its rarest parts.

We present a framework for tracking topological features directly within continuous implicit models. Such models, including implicit neural representations (INRs) and multivariate functional approximations (MFAs), are increasingly adopted to represent scientific data without the resolution constraints of discrete grids. They offer compact, smooth, and differentiable representations of complex fields, enabling new opportunities for high-performance data storage, reconstruction, and analysis. Given a continuous implicit model, our method tracks the evolution of critical points by querying the model and its derivatives, thereby eliminating the need to resample onto a grid. This approach enables faithful feature tracking while avoiding discretization-induced artifacts such as aliasing. We demonstrate the generality of our framework across a range of implicit representations, including analytic functions, MFAs, and INRs, and show that it produces smooth, coherent critical point trajectories. By enabling feature tracking directly on continuous representations, our method supports a new class of feature-driven visualization workflows centered on implicit models.

Long-horizon goal-conditioned reinforcement learning delegates control to a high-level module that proposes subgoals, but existing subgoals are implicit byproducts of value functions or latent actions, tied to the executor that produced them. We study a different object: a route-conditioned order of unavoidable stages that every successful executor must traverse, recoverable from offline trajectories and belonging to none of them. Its defining properties are topological: an unskippable stage is a separating set that every admissible path must cross, and a loop in free space forces a route choice. We read the two by homology in dimensions 0 and 1 over a transport-weighted carrier built from successful trajectories, yielding an enumerable gate set with shell-level certificates; the certified gates are what we call topological necessities. Certified gates enter the decision loop as a recursive topological gate hierarchy. Under a fixed, isomorphic free space, the object survives executor replacement: gates frozen on PointMaze data transfer without retraining to Ant and Humanoid, attaining the highest Humanoid aggregate under a unified interface (96.1), with +36.0 over a map-privileged reference on the multi-route task (p=1.4e-5); the planner saturates PointMaze (100+/-0) and matches or exceeds the strongest baselines on AntMaze (giant +22.9) and Kitchen (+15.8/+12.6).

Spatial reasoning depends not only on metric properties such as distance, angle, and shape, but also on topological relations that remain invariant under continuous deformation. Cognitive science identifies these relations as foundational to spatial understanding, yet foundation-model evaluations largely focus on metric or viewpoint-dependent relations. We introduce MindTopo, a benchmark of topological intuition across five properties grounded in cognitive science and formal topology: continuity, separation, order, enclosure, and knots. MindTopo evaluates each property at two cognitive levels. Reasoning asks a model to identify topological relations or infer how they change. Planning instantiates a foundation model as a closed-loop agent whose policy selects environment actions. MindTopo contains 11,030 instances across 13 procedurally generated task types with controllable difficulty. We benchmark 14 MLLMs and study agent configurations augmented with image and video generation, including 3 video generative models in planning settings. Every MLLM performs better on reasoning than on planning, and the best-performing model remains far below observed human performance. On Qwen3-VL-2B-Instruct, supervised fine-tuning and reinforcement learning improve reasoning more than planning. Generated observations retain local cues and reach plausible endpoints, but audited rollouts do not reliably follow environment dynamics or preserve topology across transitions. Our website is at https://mind-topo.github.io/

Closed kinematic chains complicate modular modeling by coupling active and passive coordinates through nonlinear closure constraints. This paper presents a Path-Assembled Closure Differential Mapping (PACDM) framework for modular closure resolution and kinematic reduction. Each closure element compares two ordered transformation paths with common endpoints, with their mismatch expressed through the logarithm on SE(3) and the corresponding Jacobian assembled from local transformation derivatives. Multi-path modules are constructed from a minimal set of pairwise closure elements, while rank-revealing analysis selects locally independent scalar constraints. A defect homotopy recovers closure-consistent passive coordinates from approximate estimates along a feasible and regular continuation path. At regular configurations, implicit differentiation yields the local active-to-passive differential mapping, which is subsequently used in a predictor-corrector continuation procedure for prescribed motion. The framework is evaluated on a seven-degree-of-freedom heavy-duty manipulator containing two-path and three-path closed-chain modules. Comparison with Simscape Multibody yields trajectory root-mean-square errors below 8.5 x 10^-10 rad, while predictor-corrector continuation is approximately 45.8 times faster than applying defect homotopy at every trajectory sample.

Recent advances in high-resolution fluorescence and electron microscopy have enabled nanoscale imaging across increasingly large brain volumes, but the resulting terabyte- to petabyte-scale datasets make complete neuronal reconstruction prohibitively expensive in computation, data movement, and manual proofreading. Here, we present SkNeXt, a topology-first framework for scalable neuronal reconstruction from large volumetric microscopy datasets. Instead of densely processing entire image volumes, SkNeXt first converts neuronal morphology into compact SWC skeletons that preserve long-range connectivity. Proofreading is therefore focused on sparse neuronal trees, allowing branch, continuity, and connectivity errors to be corrected before high-resolution reconstruction. The corrected skeletons then serve as persistent structural priors for recovering detailed morphology while preserving neuronal identity and topology. Crucially, SkNeXt also uses neuronal skeletons as spatial indices for selective data access, retrieving high-resolution image regions only along reconstructed trajectories and bypassing most background and signal-free volumes. This substantially reduces I/O and computational overhead, allowing reconstruction cost to scale with neuronal morphology rather than total dataset size. Using SkNeXt, we reconstructed neurons from a petabyte-scale super-resolution fluorescence dataset of the mouse brain on a single GPU within one week, without requiring exhaustive dense inference across the complete imaging volume.

Class disentanglement (the separation of a representation's class-conditional point clouds along depth and over training) is usually read off descriptive curves. We measure it as certified topological interaction between labeled point clouds, using the recently introduced Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, computed by one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test: exact permutation tests in both directions, a guarded separation certificate, and a paired test for the comparative claims applications make. Across 111 trained networks and 52,650 certified measurements, disentanglement is depth-graded and concentrated in the first epochs, and interaction quotients rank class pairs by confusability (Spearman rho=0.83), on par with cheap separability statistics. In a 96-model factorial population, augmentation is the one training choice that separates classes relative to chance; weight decay compresses the overlap without separating, and depth and width do nothing. The structural finding is one only a k-fold statistic can pose: the joint entanglement of a class triple sits below that of its strongest pair in 97% of triple-layer cells and 99.5% of deep cells, far below a measured null floor, in vision encoders and frozen language models alike. This pairwise dominance is a regularity, not a law: expected from the nesting of overlaps but not forced by geometry, present at initialization and in raw pixels, and manufactured in the last stage alone when a network memorizes random labels. The unnormalized profile mass predicts test accuracy (R^2=0.94), the quotient does not, and neither beats a linear probe. One lesson is reported in full: the paired test must use a scale-free statistic, or it certifies feature-norm dynamics as disentanglement.

Automating filament tracing in Cryo-Electron Microscopy (Cryo-EM) is essential for 3D helical reconstruction but challenged by intersecting topologies and extremely low Signal-to-Noise Ratios ($\text{SNR} = σ_s^2/σ_n^2$ < 0.1 or -10 dB). Existing paradigms fail: pixel-wise segmenters suffer from severe topological fracturing, box-based detectors face ghost center drift, sequential trackers derail due to error accumulation, and traditional active contours collapse under artificial closed-curve constraints. To resolve these bottlenecks, we present FPicker, the first topology-guided framework reconciling these incompatibilities. It unifies perception via a center-endpoint representation and an open-curve evolution module to explicitly model non-cyclic connectivity. On simulated benchmarks, FPicker outperforms top baselines by over $40\%$ relative gain in mean spatio-angular precision (mSAP) and reduces topological gap rates by over $60\%$ under extreme noise ($-20\text{ dB}$). By learning intrinsic physical geometry rather than local texture, FPicker demonstrates strong potential as a resilient geometric backbone. Its zero-shot performance on the real-world EMPIAR dataset exhibits robust topological resistance, achieving a state-of-the-art 82.9% mSAP upon fine-tuning. Our results also suggest modeling physical priors is a highly robust path toward bridging the sim-to-real gap in signal-starved scientific imaging. The code is publicly available at: https://github.com/tomzhaosky/FPicker.

Urban region embeddings have shown promising results in diverse urban sensing tasks such as crime, income, and service-call prediction. Recent methods improve representation quality by integrating mobility data with auxiliary modalities, using cross-view attention or contrastive objectives to align heterogeneous features into a unified region representation. However, leveraging the temporal dynamics of human mobility remains under-explored. Regional inflow and outflow fluctuate throughout the day, and inter-region connections emerge, persist, and dissolve over time. Moreover, prevailing fusion strategies combine views additively and miss the joint signal that emerges only when views co-occur. To address these gaps, we propose Mobility Stream-Structure Synergy (MoSS), which derives complementary views from mobility data: a Sequence view that preserves each region's hourly inflow/outflow profile, and a Structure view based on zigzag persistence diagrams that capture how regional connectivity emerges, persists, and dissolves over time. A synergy module then extracts emergent representations from the co-occurrence of these views through multi-degree interactions, explicitly capturing higher-order signal across views. Extensive experiments on New York City and Chicago show that MoSS achieves state-of-the-art performance across three downstream tasks using mobility data alone, outperforming baselines that rely on auxiliary modalities.

As AI-code assistant tools become widespread, automatic assessment of the correctness of generated code becomes a significant challenge. Code LLMs are prone to hallucinations, which may lead to code that does not solve the required problem, or even to code with severe security vulnerabilities. In this paper, we introduce CodeTD -- the first approach to pre-execution assessment of code correctness based on topological data analysis (TDA) of Code LLMs' attention maps. Our method quantifies prompt-generation mismatch using topological patterns of attention maps. We carry out experiments with common benchmarks (HumanEval, MBPP, BigCodeBench, MultiPL-E), 5 programming languages and 10 Code LLMs of size up to 34B parameters. The experimental results show that the proposed method outperforms recent baselines. Moreover, CodeTD is transferable between coding benchmarks.

Large language model (LLM) agents are increasingly proposed as autonomous SOC analysts, but two limitations make them unreliable at enterprise scale: a finite context window cannot hold a multi-thousand-host authentication graph, and free-form generation offers no guarantee that a recommended containment action is consistent with the topology it operates on. We present Sentinel-RL, an agentic-SOC architecture that decouples topological reasoning from semantic reasoning: a heterogeneous graph attention encoder summarizes the live authentication subgraph into a fixed-dimensional state, a Proximal Policy Optimization (PPO) policy maps this state to a constrained set of investigative actions, and an LLM agent loop is restricted to consuming the policy's recommendations and producing analyst-readable narratives gated by a critic. We instantiate the system on the LANL Comprehensive, Multi-Source Cyber-Security Events dataset and the Indiana University Quartz HPC cluster, reporting four results: (i) a two-phase CREATE ingestion pattern loads a 24M-edge authentication subgraph into Neo4j in 14.2 minutes on a single 32-core node, roughly 24x faster than the canonical MERGE-based pipeline; (ii) a sliding-window alert engine reliably trips a 25-event/10-second threshold in <=2.5 s across 50 trials; (iii) PPO training over 200 iterations converges to a mean episodic return of 8.74+/-0.31, with held-out precision of 0.91 and recall of 0.87 on labeled red-team events; and (iv) the integrated containment loop completes a full detect-investigate-recommend-human-approve cycle in a median of 6.3 s. We contribute a reusable engineering pattern (the hot-node deadlock workaround), a portable HPC deployment pattern (anchor-node co-location), and an enterprise-readiness analysis covering false-positive economics, reversibility guarantees, audit compliance, and the human-approval boundary.

We use tools from nonlinear algebra to study the equilibria of small linear translational spring networks. Specifically we use the techniques of homotopy continuation, monodromy, and parameter homotopy (a.k.a. cheater homotopy) to solve all rigid linear translational spring networks up to $5$ nodes in both $2$ and $3$ dimensions. We describe a method of implementing parameter homotopy that arises naturally from the physical structure of the system. We give precise total degree bounds on the maximum number of solutions for general planar spring networks. We discuss further efficiency gains obtained from polyhedral homotopy methods. We compare the computation efficiency of these techniques against a baseline of Newton's method.

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.

This paper introduces the Sierpiński-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted $d_{\mathrm{SK}}$, maps diagram points and their diagonal projections to the unit interval via the Sierpiński-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in $O(N\log N)$ steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical $2$-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted $W_Γ$, is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of $d_{\mathrm{SK}}$ over state-of-the-art approximations of $W_2$ is $626\times$, while the aggregate speedup over the full benchmark is $2100\times$. Average-linkage partitions obtained from $d_{\mathrm{SK}}$ and $W_Γ$ each exactly match the corresponding $W_2$ partition on 8 of the 12 collections. Hilbert $k$-means and Gaussian spectral clustering, both based on $d_{\mathrm{SK}}$, achieve mean adjusted Rand indices (ARI) of $0.756$ and $0.800$, respectively, with respect to the benchmark reference partitions, compared to $0.750$ obtained by average linkage on $W_2$. The Gaussian $d_{\mathrm{SK}}$ kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.

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.

Recovering compact explicit solutions from neural approximations is challenging when imperfect teacher data guide symbolic topology search and coefficient estimation. We present DeSyR, a decoupled symbolic recovery framework for differential equations. A physics-informed neural network guides repeated searches to construct candidate topologies with provisional constants. Once a topology is fixed, its coefficients are refined solely from the governing equation and prescribed constraints, followed by gated selection and verification. For linear fixed-topology parameterizations, we characterize teacher-error inheritance and show that finite-weight mixed data--physics fitting retains an $O(β^{-1})$ teacher-dependent contribution when the teacher error projects onto the model space. Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers exact coefficients; for nonlinear parameterizations, the corresponding guarantees are local. DeSyR is evaluated on 15 differential-equation problems across 18 configurations covering high-order, space--time, multidimensional, nonlinear, and coupled systems. A candidate-level audit yields a 99.23% convergence rate among free-parameter refits, while every selected refinement involving free coefficients converges. Configuration-level median refined relative $L_2$ errors are $2.31\times10^{-14}$ or lower. In same-topology comparisons, refinement reduces error by eight to fourteen orders of magnitude. These results show that an approximate neural teacher can guide topology discovery without imposing its error scale on final recovered coefficients, provided a target-capable topology is retained and physics-only refinement converges.

Deep networks trained on noisy labels simultaneously generalize on clean data and memorize flipped labels. These are usually conflated as pressures on one capacity. We present Topo^2, a measurement framework that makes them causally separable, measurable, and law-governed. Persistent-homology H1 structure of the representation space separates into a within-class manifold channel (a function of the training stopping point) and a cross-class channel (a monotone readout of memorized flipped samples). An intervention, the FM0 prescription (zero loss on flipped samples from epoch 0), reaches each setting's generalization ceiling while memorizing essentially nothing. Within the framework we establish a law set with graded evidence: (L2) FM0 separation prescription (9/9); (L1) the within-channel as a training-position function (mid-rise 6/6; convergence-back CIFAR 3/3, SVHN 2/3); (L3) a ring-construction identity (definitional, not a law); and TLS (memory-generalization topological layering): memory is causally additive, anchored (silencing clean collapses the representation), invertible (stripping memory restores near-ceiling generalization), and quantitatively billable (the memorization cost law, effective slope coefficient C ~ 0.38 at the reference capacity: CIFAR-10 0.3801 / SVHN 0.3806 / CIFAR-100 0.384 / VGG 0.3715, capacity-dependent in general and traced to clean-sample feature displacement). We also publish the framework's boundaries: a falsification ledger of nine dead ends, and an instrument-vindication section that excludes six families of global statistics as explanations of the within-channel. The framework turns "memorization" from an ill-defined capacity into a measurable, separable, invertible topological layer.

Existing GNN quantization methods suffer from considerable quantization overhead, which severely limits their practical usage in real-world scenarios. To this end, we present TopGQ, an accurate post-training GNN quantization framework, alleviating redundant quantization overhead. We propose dual-axis scale absorption, which enables activation quantization along both the outer and inner dimensions by merging one into the adjacency matrix. On top of that, we introduce TopPIN, a proxy for nodes' local structure, and use it to group nodes with similar topology during quantization. Experimental results show that TopGQ reduces quantization time by an order of magnitude while preserving accuracy.

A quantitative equational theory $U$ reasons about terms that agree up to a numerical error. It presents a free algebra $T_UA$ over a metric space $A$ of generators, the terms of the syntax at the least distance the axioms derive, and that metric is its semantic content. We give a functorial invariant of it, the persistent magnitude homology of $T_UA$: a barcode where the module is tame, finite linear algebra where $T_UA$ is finite, Lipschitz in each degree. Magnitude homology is graded by length and knows nothing of persistence, its persistent refinement nothing of where its bars begin and end, yet the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)δ$ in degree $n$ under a perturbation of size $δ$, and a computed perturbation shows that the factor cannot be dropped. An inclusion of theories induces a morphism of the presenting monads and, where the induced map is bijective and shortens no distance by more than $δ$, a comparison of barcodes under the same bound, so a barcode movement measures the metric-semantic strength of the added axioms. Four examples are computed, one in every degree.

Analog circuit design is a time-consuming, iterative process in a nonlinear and high-dimensional design space that relies heavily on expert intuition. Among recent developments, LLMs have introduced a promising approach by bringing natural language reasoning to circuit design tasks. The majority of conventional LLM-based approaches provide fragmented solutions that focus either only on sizing or topology generation. These methods require adding specific technical knowledge manually, which is inefficient and prone to hallucinations during circuit sizing. Moreover, the inherent trade-off in meeting different specs makes current approaches iterative and tedious. Another shortcoming is the inability to create innovative topologies, which may lead to sub-optimal designs due to reliance on conventional topologies. In this paper, we present AaLLM, an open-source end-to-end multi-agent LLM workflow that takes user specs as input and outputs the appropriate netlist, encompassing both topology generation and circuit sizing. AaLLM automates the creation of a relevant knowledge base from research papers and textbooks to combat tedious manual data collection. A RAG model is implemented to emulate circuit design expertise using this knowledge base. Moreover, AaLLM uses a novel tri-agent feedback system comprising a Designer that determines circuit component values, a Critic that scrutinizes these values, and an Evaluator that minimizes circuit sizing iterations by arbitrating between the other two agents. AaLLM-generated novel topologies achieve a figure of merit (FoM) comparable to that of known topologies, and up to 3x higher for certain circuits. Testing on several circuit topologies, our results show a 3x - 4.5x decrease in the number of SPICE calls at inference when compared to SOTA multi-agent LLM pipelines. The results also show a 40x decrease in wall-clock time compared to existing approaches.

Accurate foreground masks can still form an incorrect surgical-instrument instance set: duplicate, fragmented, merged, missed, or empty-frame predictions may preserve favorable pixel overlap while violating object identity and count. Final query selection is therefore a relational, variable-cardinality problem rather than a collection of independent candidate decisions. We evaluate topology-aware query selection, which represents the nonempty candidates of a fixed Mask2Former as a complete graph, learns relational candidate and pair representations, predicts set cardinality, and solves an exact structured subset problem. The formal comparison is the complete relational path versus a node-feature-matched path; it evaluates the combined effect of pairwise geometry, message passing, and the additional relational-path capacity, not an isolated component. On the sealed 22-case source test, all three discovery seeds supported instance-set performance improvement with segmentation fidelity and predefined technical-safety preservation: instance F1 increased by 0.0504--0.0612 and positive-frame set-failure rate decreased by 0.0848--0.1060. Direct ROBUST-MIPS transfer reproduced the complete result in all three seeds. Endoscapes supported only one of three seeds and therefore did not establish stable direct transfer. Taken together, the results support a bounded conclusion: the evaluated complete path improved coherent instance-set construction from fixed Mask2Former candidates in specified native-instance contracts, while stable cross-domain transfer and component-specific effects remain unestablished.

In motion planning algorithms for tethered mobile robots, the entanglement state of the tether is a critical aspect to consider during the planning phase. This is particularly important in case of a slack tether, where the shape of the tether is not determined solely by the geometry of the environment and the location of the obstacles, but also by the dynamics of the tether, by the trajectory followed by the robot, and possibly by exogenous forces. In this scenario, preventing entanglement requires planning a robot trajectory that accounts for the entanglement definition and for the dynamics of the robot and of the tether. In this work, we propose a motion planning algorithm for tethered mobile robots with a slack tether that computes dynamically feasible entanglement-free trajectories to navigate through an environment with static obstacles. By considering the entanglement state during all the stages of the planning pipeline, we are able to compute safer trajectories that avoid entanglement during the motion of the robot. We achieve this through a three-step pipeline, which includes (i) the construction of a topological model of the entanglement-free configuration space of the tethered robot, (ii) the generation of a set of candidate paths using this model, and (iii) the computation of a dynamically feasible entanglement-free trajectory by solving a homotopy-constrained trajectory generation problem. The resulting trajectory can then be executed to lead the robot to its target location, while maintaining the tether in an entanglement-free configuration. We demonstrate the benefits of this algorithm in simulations, where we show how the planning algorithm avoids violations of the entanglement constraints, resulting in safer and more reliable trajectories.

Electrical circuit analysis requires more than recognizing components in an image. A solver must ground symbols and labels, recover latent topology, select a physical model, formulate coupled equations, propagate intermediate quantities, and preserve units, signs, directions, and phase conventions. We introduce \benchmark, a benchmark of 1,000 authentic textbook problems for evaluating this complete long-horizon visual-to-symbolic reasoning process. Each problem pairs one or more circuit diagrams with a self-contained question, a typed or semantically specified answer, and a reference worked solution. An evidence-first construction pipeline aligns questions, figures, and solutions, while a reasoning-oriented taxonomy organizes problems by circuit type and dependency depth. Evaluation combines conservative typed scoring with identity-blinded multi-model semantic consensus, retaining every problem in the denominator. Across three commercial chatbot systems and six open-source multimodal large language models, the highest-scoring system reaches 84.8% accuracy. However, performance consistently deteriorates on long-horizon problems, and qualitative analysis exposes persistent failures in topology-to-target binding, physical conventions, and late-stage output propagation. \benchmark{} provides a focused testbed for measuring whether multimodal models can transform technical visual evidence into sustained, physically valid symbolic reasoning. Code are available at GitHub - CircuitReason/CircuitReason1K.

Hallucinations are a major concern for the integration of artificial intelligence into medicine, although less explored in the realm of medical image processing. Unlike problems in natural text understanding and reasoning therewith, determining whether or not predictions derived from biomedical images and signals is less intuitively clear. This article suggests that topological errors could constitute hallucinations in a way that can be more readily measured and thus regulated. Certain of these properties for certain types of problems, such as biomedical signal segmentation, can be rephrased as linear temporal logic predicates, a number of which can be explicitly enforced using probabilistic graphical models. Our simulations show the potential of these explicitly constrained predicates for the case of automatic surgical phase recognition in robot-assisted hysterectomy, improving accuracy by approximately 10% while removing the vast majority of topological errors, suggesting that mathematical guarantees of correctness can supplement other empirical forms of regulating machine learning in medical image computing and computer-assisted interventions.

Current LLM safety guardrails face a fundamental tension: fine-tuning distorts pre-trained representations while generative judges incur prohibitive inference costs. We ask a complementary question: how far can safety be achieved through pure geometric reasoning over frozen representations, and where does it fail? We present HoloAegis, a minimally parametric topological inference framework that decouples representation from reasoning: an un-fine-tuned encoder maps text to the unit sphere S^{d-1}, and all decisions reduce to Gibbs-Boltzmann free-energy differences over pre-computed anchor centroids. We contribute a boundary-mapping study rather than a leaderboard claim. On a frozen three-benchmark protocol, HoloAegis (3.2 MB) statistically matches WildGuard-7B (14 GB) on toxicity (0.96 vs. 0.96), exceeds it on harmful behaviors (0.99 vs. 0.79), and cedes oversafety detection (0.62 vs. 0.98) -- while ShieldGemma-2B fails on indirect harms (0.34). These failure modes are complementary and mechanistically traceable: potential-difference scoring senses manifold clustering, whereas policy-conditioned LLM judging requires explicit taxonomy matching. We restate our Topological Boundary Stability conjecture in ratio form and validate it via reference-set bootstrap: anchor banks reduce score variance 4-15x and boundary displacement to approximately 0.44 + 0.23 sqrt(k/K) of the full-space estimator. Per-domain analysis further reveals that geometric separability tracks within-domain semantic homogeneity. Our results chart where geometric guardrails substitute for, and where they must defer to, LLM judges.

Recent years have witnessed the advanced development of topology inference research, which helps elucidate the interaction relationships of components in many biological networks. This paper focuses on inferring the topology of a group of immune cells, based on the collected data from cell-depletion based experiments. The problem is very challenging due to i) the lack of standard analytical models for the cell interactions, and ii) the restrictive data availability determined by the huge experiment and time costs. To address these issues, we first leverage certain common knowledge and observations on the experiments to characterize three properties on the cell amounts during the interaction process: state non-negativity, ratio-based convergence, and triple signs of topology weights. Then, we construct a new model with simple structure and analytical convenience, and obtain sufficient conditions for the model to accommodate all three properties. Finally, based on the constructed model, we propose a constrained quadratic programming method to infer the topology from limited number of data pairs. Validation on experiment data demonstrate the effectiveness of the proposed method.

Transformers have had a profound impact on the world of language processing and computer vision. As efforts to answer the million-dollar question of ``How does a Transformer learn?" have been increasing, existing interpretability studies primarily analyze representations at isolated layers or the network as a whole, while the developmental evolution of individual representations and its manifolds across transformer layers remains underexplored. With this work, we aim at providing a comprehensive analysis of the evolution of representations as the representation point cloud transforms across the layers; thereby attempting to isolate layers or establish a trend which comes closer to justifying how and when raw input representations evolve into task-relevant feature representations. Thus, Transformer Geometry Observatory-TGO-IV introduces a topological framework for analysing the evolution of Transformer representations through the lens of Persistent Homology. Rather than studying local geometric properties alone, TGO-IV constructs Vietoris--Rips simplicial complexes from token-level representation point clouds and investigates the evolution of their persistent topological signatures across Transformer layers. The proposed framework comprises complementary topological observatories including Persistence Diagrams, Barcode Diagrams, Betti Curves, Persistence Landscapes, Bottleneck Distance, and Wasserstein Distance, enabling a comprehensive analysis of how the global topology of representation point clouds develops throughout the forward pass.

Accurate estimation of building energy use is essential for achieving carbon neutral and sustainable buildings. To better understand the influence of design decisions on building energy use and calibrate machine learning models that can give architects and engineers rapid design feedback, large-scale datasets are needed that explicitly map building geometry to performance. We present ArchEGraph, a large-scale benchmark dataset that represents buildings as heterogeneous graphs with aligned geometry, topology, weather, and zone-level thermal loads. The dataset contains 5,481 buildings and 49,326 validated building-weather simulation cases. In total, it includes over 133,000 space nodes and 1.44 million face nodes, reflecting substantial geometric and topological complexity. Based on ArchEGraph, we define two benchmark tasks: (i) graph reconstruction from polygonal meshes, aiming to recover topological structure from geometric representations; and (ii) topology-informed load prediction, which leverages graph structure and temporal weather conditions to forecast zone-level response time series. We further introduce standardized evaluation protocols for both tasks and conduct cross-building and cross-climate generalization experiments to assess model robustness. ArchEGraph provides a unified testbed for studying geometry-topology-physics coupling in building energy modeling, enabling the development and evaluation of scalable and generalizable surrogate models.

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

Dynamic graph prompting freezes a pre-trained temporal backbone and adapts it to label-scarce downstream tasks using lightweight prompts. However, existing methods operate within a single, fixed embedding space. In this work, we reveal that temporal shifts in local clustering and degree heterogeneity actively reorganize the edge curvature spectrum---indicating that the optimal representation geometry dynamically evolves with local topology over time. We formalize this unaddressed mismatch as geometry under-adaptation. To overcome this limitation, we propose CurvPrompt, a topology-routed geometry prompting framework for dynamic graphs. Instead of relying on a single space, CurvPrompt maintains a bank of curvature-diverse Riemannian experts, each paired with a learnable prompt. A topology-aware gate dynamically routes each node--time instance to a sparse subset of experts, constructing a personalized mixed-curvature representation. To ensure parameter efficiency and training stability under extreme label scarcity, CurvPrompt employs soft routing during pre-training to build a continuous topology--geometry mapping, and transitions to hard Top-K routing with uniform weights during downstream adaptation. Extensive experiments across four benchmark datasets show that CurvPrompt significantly advances few-shot link prediction while delivering strong, consistent performance on node classification tasks, validating the necessity of geometry-adaptive prompting.

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.

Particle swarm optimization (PSO) has been widely applied to solve complex optimization problems from real-world applications due to its efficient exploration of large solution spaces and the ability to converge towards optimal solutions without requiring gradient information. Common swarm topologies in standard PSO and its variants, e.g., Ring and Star, can be regarded as graphs, where each edge connects only two particles. Such topology structures allow direct interactions only between connected particle pairs, and thus often fail to directly capture the higher-order social relationships that are necessary for navigating complex search landscapes. Therefore, this article proposes a novel PSO variant termed Hypergraph-assisted Particle Swarm Optimization (HPSO). In HPSO, the topology of the particles in a swarm is modeled by a hypergraph, in which hyperedges are used to connect multiple particles. This allows multiple particles within a hyperedge to interact directly. Furthermore, an adaptive hypergraph updating strategy is designed to periodically reconstruct the topology based on cumulative average particle displacement, thereby maintaining swarm diversity throughout the evolutionary process. In the experiments, the effectiveness of HPSO is verified on the IEEE CEC'17 benchmark suite, and the results demonstrate that HPSO achieves promising performance across various types of functions. Furthermore, the ablation experiment demonstrates that HPSO has excellent search capabilities.

LLM-based multi-agent systems (MAS) typically optimize a single topology, restricting reasoning to a narrow trajectory and limiting comprehensive analytical capacity. Naively merging multiple topologies into a composite graph introduces redundant noise propagation across irrelevant connections, degrading solution quality. To address this dilemma, we propose \textbf{Hierarchical Sparse Coordination over a Union of Complementary Topologies for MAS (HELENA)}, a multi-agent framework that balances diverse reasoning paths with sparse task-dependent execution. \helena{} constructs a union MAS graph from complementary candidate topologies selected via Monte Carlo Tree Search and Determinantal Point Process, broadening the reasoning trajectory for comprehensive analysis of complex problems. A Hierarchical Sparse Coordination module then activates only a sparse subgraph at each step while agents exchange compressed latent briefs to suppress redundant noise propagation. Finally, a Local Self-Refinement stage identifies decision units with discrepancy evidence and rewrites them only when contrastive evidence simultaneously confirms a reliable solution-side failure and a challenger-side improvement. Experiments across eight benchmarks show that \helena{} achieves state-of-the-art results on all benchmarks, with an average gain of \pctup{3.47} over the strongest baseline and up to \pctup{10.34} on MMLU-Pro, achieving larger improvements on harder benchmarks at a reasonable additional cost.

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.

Developing robust flood assessment models requires high-quality paired satellite imagery, yet such data remain scarce for flood-specific image generation. Although generative models provide a promising means of data augmentation, existing methods often yield implausible spatial layouts of flooded regions and distort scene structures. We propose FlowForm, a framework for satellite flood synthesis that integrates SWE-inspired latent regularization with structure-aware conditioning. The Flood Descriptor Module (FDM) imposes differentiable penalties on residuals of the steady-state Shallow Water Equation in auxiliary latent fields at the diffusion bottleneck. The Terrain Anchor Adapter (TAA) injects depth, semantic, and edge features at four encoder scales of the U-Net. We further curate FloodScape, a large-scale, high-resolution dataset comprising paired satellite images acquired before and after disasters. In addition to standard image-generation metrics, we evaluate the consistency of flooded regions, zero-shot generalization to a geographically held-out flood event, and sensitivity to individual components. Across all reported comparisons, FlowForm achieves higher visual fidelity, greater similarity between paired images, and stronger consistency of flooded regions.

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.

We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on MNIST ($\geq 95\%$ test accuracy), and measure how topological features change across layers for different architectures and activation functions. We find that smaller PCNs collapse connected components across layers earlier than larger models (Spearman $\unicode{x1D70C} \in [0.72, 0.79]$ across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation ($\unicode{x1D70C} = -0.58$) between the depth at which simplification occurs and reconstruction error; i.e., architectures that simplify later reconstruct better. Finally, a seed-level bootstrap comparison across architectures and activations shows that PCNs consistently collapse connected components later than matched MLPs, with an average difference of $3.6$ layers. These results suggest that persistent homology offers a useful quantitative lens on the compression--reconstruction tradeoff in PCNs, and that both model capacity and the recurrent, bidirectional dynamics of predictive coding inference shape when this tradeoff is resolved across layers.

Language model benchmarking is a difficult task. Outcome reasoning alone does not test the model's conceptualization of language and popular open-source benchmarks are quickly saturated or ingested as training data. It is important to test the model's output, but augmenting these tests by characterizing semantic structure gives more insight to how models relate abstract concepts. However, the high dimensional embedding spaces are not easy to interpret. This work demonstrates how topological methods can be used to rigorously compare these spaces to low dimensional and interpretable baselines like ontologies and curated knowledge graphs. These multi-modal alignment tests make it possible to track model adaptations and test phrase understanding across multiple languages.

Stage-one stellarator design searches a high-dimensional family of three-dimensional plasma boundaries and fixed-boundary MHD equilibria for configurations that jointly meet requirements on confinement, field-line topology, force balance, stability proxies, and geometry. These specifications do not provide a general constructive map to a validated finite-beta equilibrium. High-quality targets are commonly developed through iterative numerical optimization whose outcome depends on the initial configuration, active Fourier resolution, objective priorities, and local solver budget. Coordinating this process is computationally costly and expert-intensive, limiting both design throughput and the production of consistently evaluated data. We present a proof of concept for \emph{agentic} stage-one optimization. A bounded language-model agent diagnoses the current equilibrium and selects the next local optimization experiment, while deterministic DESC execution owns prescribed profiles and flux, symmetry, metric evaluation, solver validity, and acceptance. On a common-budget subset from an expanding finite-beta campaign, the number of gate-valid configurations increases from five inputs to nineteen outputs; median Boozer QS RMS decreases from $2.39\times10^{-4}$ to $1.07\times10^{-4}$, and median maximum principal curvature decreases from $62.56$ to $33.00\,\mathrm{m}^{-1}$. A complementary long route achieves a $9.10\times$ QS reduction while repairing magnetic-well and curvature defects. The system also records every attempted local action as transition evidence, yielding 734 structured parent--action--outcome records in the reported experiments. These results show that agentic outer-loop control can sustain finite-beta, multi-objective search and turn repeated optimization into a scalable source of improved equilibria and reusable decision data.

Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f requires polynomial tests of degree N* >= log(1/f)/(2 eps), proved via the Chebyshev extremal problem; a Gauss-quadrature construction gives N* >= 4b-1 for a b-scale topology, so cost is set by feature fineness, not feature count. The law is one-sided: we exhibit an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H_1 is nonzero. Its practical content is a calibration rule. The upper bound is attained by Gaussian test functions, the RKHS witness of an RBF kernel, so the law predicts which bandwidth an MMD test should use: the feature scale. On real embedding streams we measure sigma*/eps with median 1.12 (IQR 1.01-1.52, n=26) over three settings and three scales, and a data-driven bandwidth reaches AUC >= 0.95. Against an adversary optimised against the defender's statistics (mean, covariance, k-NN, kurtosis), only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends on choices usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR 1% where the first persistence landscape attains 0.00. What survives is a cost gap, not a power gap: where persistence works it costs 116x kurtosis, which works at least as well. We conclude not that topological summaries are useless, but that on this task a kernel test whose bandwidth the law sets dominates them.

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.

Large language model (LLM) agents are increasingly deployed in scientific research, where reliability is critical and the underlying knowledge is densely interconnected. In such settings, hallucinations are particularly damaging: a single erroneous claim on a foundational concept can propagate through multi-step reasoning and corrupt entire trajectories. Existing hallucination benchmarks largely operate at the surface level, treating facts in isolation and relying on uniform accuracy metrics that ignore this topological structure. We address this gap with SCHEMA, the first evidence-grounded, topology-aware evaluation framework for hallucinations in scientific agents. SCHEMA automatically constructs scientific concept graphs from benchmark seeds and literature evidence, synthesizes graph-grounded tasks spanning claim verification, multi-hop reasoning, open-ended explanation, and experimental code generation, and evaluates agents with two complementary diagnostics. A trajectory hallucination pipeline audits intermediate reasoning at scale via a topology-weighted severity score, while a multi-agent counterfactual attribution module pinpoints the causal mechanism behind selected failures. SCHEMA reveals that hallucinations concentrate at a small set of highly connected knowledge hubs, and that final-answer accuracy decouples from trajectory honesty; models often reach correct conclusions through structurally flawed reasoning. These results indicate that for high-stakes scientific applications, terminal accuracy alone is an insufficient signal of agent reliability, motivating mechanism-level evaluation grounded in knowledge topology. Code is available at https://github.com/circles-post/SCHEMA.

Modern opaque AI models prize performance over interpretability, which makes testing difficult. However, formal statistical tests conducted on a model's embedding space can provide robust characterizations of semantic structure, concept separation, and knowledge graph alignment. Model developers would benefit from a model comparison technique that leverages human-curated knowledge structures to test alignment. The scale of the input space for even relatively simple tasks motivates the need for alignment checks that augment standard outcome reasoning. This work develops and demonstrates a topology-based multi-modal alignment test to make deployment, selection, and comparison of opaque models more interpretable. These methods also offer an intuitive connection to possibility theory and a unified decision theoretic framework from data to deployment.

Over the last decade, neural networks have been applied to an increasingly diverse range of applications, including data with rich geometric, topological, or symmetry-related structure. As a result, researchers have increasingly drawn inspiration from topology, algebra, and geometry. Despite this rich algorithmic development, the supporting software ecosystem remains fragmented. Many important methods exist only as research prototypes in unmaintained repositories. We address this by introducing Topology, Algebra, and Geometry Torch (TAGTorch), an open-source, PyTorch-based library that unifies tools inspired by topology, algebra, and geometry, including data-preprocessing methods, architectures, training techniques, and model analysis tools. We describe the design philosophy of TAGTorch and then discuss its current architecture and capabilities, highlighting areas where it can fill gaps in the current software ecosystem. We conclude with a discussion of our future development priorities for the library.

Large language model-based multi-agent systems improve complex problem solving through task decomposition, agent specialization, information exchange, and intermediate validation. However, existing systems typically treat communication topology as a fixed design choice or an offline optimization target. We introduce MANTA, a framework for Multi-Agent Network Topology Adaptation that enables communication structures to self-evolve at inference time. Before execution, MANTA initializes a task-conditioned topology from prior structural experience. During deployment, it monitors collaboration traces and applies bounded structural updates when the current organization becomes insufficient. These updates can modify agent roles, communication links, execution order, information visibility, and validation pathways while preserving the task interface and agent budget. We evaluate MANTA against representative single-agent and multi-agent baselines on five benchmarks spanning information seeking, tool use, planning, workflow execution, and mathematical reasoning. MANTA achieves the highest average score of 74.0, outperforming the strongest baseline by 5.8 percentage points and obtaining the best result on PlanCraft. These results show that inference-time self-improvement can extend to the architecture of collaboration itself.

We introduce Topoformer, a lightweight and scalable framework for graph representation learning that encodes topological structure into attention-friendly sequences. At the core of our method is Topo-Scan, a novel module that decomposes a graph into a short, ordered sequence of topological tokens by slicing over node or edge filtrations. These sequences capture multi-scale structural patterns, from local motifs to global organization, and are processed by a Transformer to produce expressive graph-level embeddings. Unlike traditional persistent homology pipelines, Topo-Scan is parallelizable, avoids costly diagram computations, and integrates seamlessly with standard deep learning architectures. We provide theoretical guarantees on the stability of our topological encodings and demonstrate state-of-the-art performance across graph classification and molecular property prediction benchmarks. Our results show that Topoformer matches or exceeds strong GNN and topology-based baselines while offering predictable and efficient compute. This work opens a new path for parallelizable and unifying approaches to graph representation learning that integrate topological inductive biases into attention frameworks.

Persistence Diagram (PD) is known to capture point cloud topology effectively, but its computation has high time complexity. Expected Persistence Diagram (EPD) has been developed to reduce the time cost by studying the topology of multiple subsets of a point cloud and it serves as a distribution of topological features. Existing EPD vectorizations often rely on predefined point transformations, such as Gaussian or landscape functions. We study an alternative discretization based on Voronoi histograms, which trades smooth functional approximation for adaptive partition-based counting. We propose to use Voronoi Diagram-based histogram as the vectorization of EPD, without imposing an explicit smooth point transformation model. Under stated separation and normalization conditions, we establish stability bounds and characterize when the histogram representation preserves Wasserstein-scale variation. We demonstrate the effectiveness of our proposed representation on real-world datasets which have significant topological features for classification and dimensionality reduction tasks.

Multi-agent planning problems arise in a variety of engineering applications, such as multi-robot wildfire fighting and unmanned aerial inspection in factories. A particular challenge is the existence of spatio-temporal (i.e., when and/or where an agent should do what) and topological constraints (i.e., how agents should interact), as typically formalized via the notion of graphs. Over the last years, various frameworks have been proposed that can capture such constraints via spatio-temporal logics. We focus here on spatio-temporal logic with graph operators (STL-GO), a recent formalism that supports reasoning about multiple agents and their topologies, such as sensing, communication, and task topologies. In this paper, we consider the problem of planning multi-agent paths that satisfy constraints written in STL-GO. This problem is particularly challenging due to the need of encoding multiple, potentially time-varying graphs via the graph operators inherent to STL-GO. We present two encodings of this problem, one based on mixed-integer programming (MIP) and another based on satisfiability modulo theory (SMT), with soundness guarantees. We provide a unified interface for specifying agent constraints, their graph topologies, and the STL-GO specification, enabling seamless use of both methods and facilitating direct comparison between them. We evaluate both encodings on a multi-UAV search-and-rescue benchmark, ablating over team size and graph complexity, highlighting the expressiveness of the proposed encodings under dynamic multi- graph interactions.

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.

Enzyme function prediction is a hierarchical, knowledge-intensive form of protein function classification. Existing benchmarks expose an anomaly: general LLMs often get the coarse first level right, yet once asked for a complete EC number their accuracy at levels two through four drops to almost zero, while specialized models and tools stay usable. We propose EC-Reason-Bench, a training-free, diagnostic evaluation protocol built to answer two questions: why general LLMs score close to nothing on EC number prediction, and how much of that loss can be recovered without updating a single weight. We break enzyme classification ability into four orthogonal levers that can each be measured on their own: output structure, external knowledge, reasoning structure, and reasoning robustness. We test each lever with an inference-time method against a shared zero-shot baseline reproducing previously reported near-zero performance. Experiments with several strong reasoning LLMs yield four main findings. First, external knowledge is decisive and must precede reasoning: uniformly low closed-book performance rises sharply with open-book access, narrowing model gaps. Second, in closed-book settings, whether cascading and chain-of-thought help or hurt depends on a model's tendency to abstain. Third, once evidence is available the aggregate score of the best LLM setting is indistinguishable from simply voting the EC numbers of the nearest retrieved neighbors; that tie is an artifact of averaging, and it hides a large gain on adversarial evidence set against an equally large loss on multi-functional enzymes. Reasoning over evidence therefore acts as an arbiter of conflicting neighbors rather than as a source of knowledge, and no single-number leaderboard can see it. Fourth, accuracy obeys a law of homology availability.

Kubernetes is central to the cloud-native ecosystem, orchestrating containerised workloads. Recent work suggests that large language models (LLMs) can automate cluster security remediation, generating configuration patches from Kubernetes Security Posture Management (KSPM) findings without human authoring. Such systems, however, prompt the model with each finding in isolation from the live service call graph, assuming general hardening knowledge suffices. This assumption breaks down whenever a patch must preserve a runtime service dependency invisible to the model: an otherwise compliant fix then carries a destructive functional blast radius, crashing downstream callers or silently severing call edges across the cluster. Whether live cluster context improves patch correctness has not been measured under controlled conditions across multiple dependency classes. We introduce KuTIE (Kubernetes Topology Intelligence Engine), which builds a live cluster context from Istio call edges, Trivy KSPM findings, and the service-account bindings a workload reads, and conditions LLM patch generation on it. It is evaluated on VulnCare, a purpose-built 36-deployment, four-namespace healthcare cluster with 31 injectable findings across seven dependency classes, each labelled by topology dependence against cluster ground truth. Across 248 trials, topology context raises topology-dependent patch correctness from 11.1% to 78.0% ($Δ= 0.669$), a gap that holds for every model and for six of seven classes, from credential and network-policy ($Δ= 0.95$) to role-based access control ($Δ= 0.31$); a topology-independent control exhibits no such effect ($Δ= 0.0$), isolating the result from generic prompt enrichment. Supplying the live service-call graph and the service-account bindings it exposes thus improves remediation of topology-dependent findings well beyond scanner-only context.

Polygonal meshes are the standard surface representation of modern 3D pipelines, and generating high-quality meshes with artist-style topology is essential for film, gaming, and interactive 3D applications. Mainstream approaches serialize a mesh into a token sequence and decode it autoregressively, which is slow at inference and sensitive to error accumulation, making them impractical for interactive asset creation. We present Meshy T2, a fast native mesh generation framework built on flow matching. At its core is a vertex-set mesh VAE that encodes a mesh into one continuous latent token per vertex and decodes vertices, edge connectivity, and face winding order in a single pass, preserving high-precision geometry and artist-authored topology without vertex quantization or welding. Generation proceeds as a coarse-to-fine cascade of two flow-matching models: an image-conditioned voxel flow first sketches the overall shape as a coarse occupancy scaffold, and a mesh flow then populates the scaffold with per-vertex latent tokens, conditioned on the image, the scaffold, and a requested vertex budget. This design delivers three practical capabilities: interactive generation speed through parallel flow-based synthesis; effective face-count control through the requested vertex budget; and native support for multi-part assets, whose components emerge directly from the generated connectivity. In our experiments, Meshy T2 achieves state-of-the-art geometric fidelity and completes end-to-end image-to-mesh generation within a median of 6 seconds, over an order of magnitude faster than autoregressive baselines. Code and weights will be available at https://github.com/meshy-dev/meshy-t2.

Contemporary machine learning struggles to learn continually, reuse prior knowledge, and expose a comprehensible internal structure. A recently proposed developmental, gradient-free learning framework addresses these limitations by learning a discrete, topological model of its inputs through local variation and selection, yielding an inherent continual-learning guarantee: new observations refine existing structure without overwriting past knowledge, and without replay buffers or predefined task boundaries. Its extension to visual inputs demonstrated this principle on shape recognition, but relied on a feature representation of limited expressivity that capped recognition accuracy. We introduce a new visual feature representation that encodes shape structure across multiple scales, capturing edge and contour features together with their spatial relations, and integrate it with the network-refinement learning process; we further improve the learning dynamics and the read-out used to predict from the learned model. The study targets two-dimensional shape, with class-incremental MNIST as a controlled, interpretable benchmark in which continual-learning behavior can be measured directly. Our approach substantially increases accuracy over the prior representation, matching or exceeding replay- and regularisation-based baselines at comparable storage while storing no past data, and preserves the framework's defining behavior: earlier-learned classes are retained as new ones are introduced, with no destructive adaptation, and the learned representations remain human-interpretable. What separates the methods is retention: the baselines surrender most of a just-trained class within its own cycle and relearn it afterwards, which ours does not. The significance lies in the manner of learning. The system integrates information one sample at a time while provably preserving its responses to...

In practical settings, medical image segmentation models are often developed with limited annotated data rather than fully labeled datasets. Training frequently begins in ultra-low labeled regimes where only a small number of volumes are annotated. In such scenarios, practitioners must simultaneously decide which cases to annotate and how to best use the remaining unlabeled data. Although active learning (AL) and semi-supervised learning (SSL) both target annotation scarcity, they are typically designed and optimized independently, resulting in objective mismatch and unstable training during early-stage "cold start" conditions. We propose RegAL, a unified active semi-supervised framework governed by a shared topology-aware Pareto optimization that couples sample acquisition with unlabeled data utilization. RegAL evaluates images along three complementary axes, voxel-wise uncertainty, feature diversity, and a novel topological consistency metric, to select anatomically informative edge cases for annotation. On the other hand, the same criteria are used to identify geometrically stable atlas candidates for diffeomorphic registration-guided augmentation to train a self-supervised Mean Teacher segmentation network. Across BraTS 2021, dHCP, and ProstateX, RegAL remains stable with few labeled volumes and consistently outperforms state-of-the-art AL, SSL, and active semi-supervised baselines across Dice and boundary-distance (ASD, HD95) metrics under extreme annotation scarcity.