Sheaf Theory

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4 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

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Latest papers 20

Oct 1, 2026cs.AI

Global Coherence: When Every Agent Is Right and the Team Is Still Wrong - A Local-to-Global Semantic Foundation for Multi-Agent Collaboration

AI agents can each make locally valid decisions yet jointly produce an invalid result. We call this the global coherence problem: a failure of shared state, not merely of model intelligence. Our Observation-Aliasing Impossibility Theorem gives the exact boundary. A policy can guarantee a valid action exactly when all worlds producing the same observation share an admissible action. If k indistinguishable worlds require pairwise-disjoint actions, the best randomized worst-case success is 1/k; more reasoning, roles, messages, or samples cannot recover the missing distinction. A stronger model can reason better within its context, but it cannot see beyond it. We then give local-to-global runtime semantics X = (H, C, G, F; D): topology H records overlapping scopes; category C governs state-changing actions; groupoid G retains reversible translations; sheaf F tests whether local views glue into one world; and minimal history D keeps only distinctions that alter legal futures. Models propose; the harness owns shared state and governs commit. Nine studies test both the failure and its boundary. On a controlled revision benchmark, the same frontier model scores 40/40 when the deciding event is visible; when it is hidden, tested arms score 12--17/40, consistent with chance (1/3); restoring one authoritative fact returns 40/40. On TeamBench, ordinary teams exceed a shared budget in 5/5 runs, a visible live count leaves 4/5 violations, and commit enforcement leaves 0/5. In tau2-bench Telecom, current-state checks score 0.07 after silent reverts, while the harness scores 1.00. Where a conventional solver already owns the complete relevant state, it ties the harness as predicted. The counterintuitive conclusion is that local intelligence cannot substitute for missing global state.

Xin Heng

Tote AI

Multi-Agent Collaboration Sheaf Theory Multi-Agent Systems

Oct 1, 2026cs.LG

Clifford Sheaf Neural Networks

We introduce the Clifford Sheaf Neural Network (CSNN), an equivariant sheaf neural network for geometric graphs that places a Clifford algebra on each stalk of a cellular sheaf and transports multivector features along edges. The canonical choice of restriction map for sheaves with algebra-valued stalks is algebra homomorphism. Adding the constraint of equivariance, the naive choice becomes versor conjugation. However, versor conjugation is expressively weak, so we drop algebra homomorphism and arrive at the K-term sandwich. The resulting sheaf Laplacian is positive semidefinite by construction, needs no versor constraint, and still mixes grades. Our main contribution characterizes the resulting family of restriction maps along three axes: which grades a map couples, how much of the endomorphism space it reaches, and how well it is conditioned. The K-term sandwich spans half of the endomorphism space, and in Cl(3, 0, 0) it corresponds to the maps that commute with the central pseudoscalar. The number of terms controls expressivity. CSNN is the reversion member, a first-order model by construction and the grade-mixing corner of this family, developed as a sheaf construction for graph-level equivariant regression.

Kotaro Kamiya, Joel Nicholls

SyntheticGestalt KK

Sheaf Neural Networks Equivariant GNNs Sheaf Theory

Sep 14, 2026cs.LG

Space as an Interventional Invariant: Cross-Modal Predictive Geometry for Stratified Cities and Em-Spaced Intelligence

Space is a foundational concept across mathematics, physics, spatial cognition, urban science, and embodied intelligence, yet these fields often treat spatial structure either as a shared geometric container or as a collection of disconnected representations. Such approaches struggle to explain how heterogeneous sensory and urban processes can jointly reveal a common spatial structure, particularly when different modalities do not share the same metric or representation. This paper addresses this gap by defining space as an interventional invariant: the minimal relational structure that preserves local compatibility and the conditional laws of future observations under admissible actions. We develop a cross-modal predictive geometry that integrates local state spaces, modality-specific observation maps, an action groupoid, and a canonical predictive-state quotient, with explicit causal conditions for identifying interventional rather than merely observational structure. The key theoretical result shows that, under joint point separation, equivariance, and interventional faithfulness, the latent space is identifiable up to the centraliser of the intervention group, thereby reducing representational ambiguity to residual coordinate freedom. The framework is further extended to stratified urban systems using sheaf-valued representations, allowing geometric, physical, mobility, social, and economic layers to coexist without being reduced to a single metric. Synthetic experiments under noise evaluate equivariance, predictive sufficiency, holonomy, restriction-map recovery, cross-scale consistency, and context saturation. The resulting framework provides a unified and falsifiable foundation for spatial cognition, urban science, embodied AI, and em-spaced intelligence.

Tao Yang, Xuhui Lin, Kunyao Li +1

School of Architecture, Tsinghua University, Beijing, China · Department of Geography, University College London, London, United Kingdom · School of Engineering, Cardiff University, Cardiff, United Kingdom

Causal Representation Learning Geospatial Representation Learning Sheaf Theory Spatial Reasoning

Sep 8, 2026cs.AI

Time-Varying Data as Sheaves: an Invitation to Narratives

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

Wilmer Leal, Benjamin Merlin Bumpus, Jana K. Nickel +3

Department of Mechanical and Aerospace Engineering, University of Florida, 1064 Center Drive, Gainesville, 32611-6250, Florida, USA. · Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Rua do Mat˜ao, 1010, S˜ao Paulo, 05508–090, SP, Brasil. · Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, Hamburg, 20146, Germany. +2

Time Series Reasoning Sheaf Theory

Aug 13, 2026cs.AI

Capability Sheaves for Compositional Agent-Harness Repair: Controlled Quotients and a Real-Repository Stress Test

Agent harnesses combine retrieval, routing, state, provenance, and verification, but locally successful components may disagree on shared state. We model this failure with a finite \emph{capability sheaf}: stalks encode typed behavior signatures, restriction maps retain shared fields, and accepted runs are useful global sections. An exact finite constraint-satisfaction problem (CSP) defines acceptance, while a linearized relative cohomology class provides a diagnostic and search feature. A controlled experiment over 20 task clusters introduces hidden interior mediators whose raw states are nuisance variables. Quotienting their coboundaries reduces the candidate budget from 2,000 to 1,000 per cluster; aligning the hidden state removes the gap. Exact CSP matches the quotient, so the result demonstrates invariance to stale representatives, not superiority over exact reasoning. We then test the method on a discovery split from the SWE-bench Multilingual pool of PatchFuseBench: 160 issues from 20 repositories, 875 real candidate patches, 2,579 source-aware edit atoms, and 153 newly executed patches. A first pool-level construction is constant because

[b-Dx]=[b]

in

\operatorname{coker}D

and therefore cannot rank configurations. A candidate-indexed repair is nontrivial on 848/875 candidates and varies within 120/160 issues. It resolves 118 issues versus 116 for a matched noncohomological selector, but the difference is not supported across repositories (exact sign-flip

p=0.75

). A leave-one-repository-out abstention gate reaches 127/160, tying the strong anchor and exceeding its matched gate by one issue (

p=1.0

). The discovery gate therefore fails and the confirmatory split remains sealed. The study supports the controlled invariance mechanism and an identifiability correction, but not a real-world cohomological advantage.

Saveliy Batruin

Independent researcher

Automated Program Repair Sheaf Theory LLM Agent Harnesses

Aug 3, 2026cs.LG

Benchmarking Sheaf Neural Networks for Inductive Tasks

Sheaf Neural Networks (SNNs) generalize message passing by replacing scalar edge weights of standard Graph Neural Networks (GNNs) with learnable, edge-dependent restriction maps between node stalks. Despite their strong theoretical foundations and promising transductive results, SNNs have been evaluated almost exclusively on transductive node classification, leaving their behaviour under inductive protocols unknown. We address this gap through the first systematic benchmark of the sheaf design space, evaluating three diffusion mechanisms (neural sheaf diffusion, sheaf attention, and sheaf attention with Graph Attention Network v2), three restriction-map parameterizations, three stalk dimensions, and six modern GNN architectural components, within a message-passing reformulation that never assembles the heavy sheaf Laplacian, making the full design space trainable under cross-graph batching. Across

1{,}890

controlled experiments on 14 inductive datasets, multiple insights emerge: restriction maps are the dominant design choice and general maps are preferable, larger stalks add capacity but not long-range reach, architectural components explain more performance variation than the entire sheaf-specific design space itself. Under a matched protocol, SNNs transfer to inductive settings but do not reach the strongest baselines, with gaps being dataset-dependent. Practically, a single sheaf configuration can generalize across datasets, so effort is better spent tuning the surrounding architectural recipe than the sheaf operator itself.

Stefano Fiorini, Edoardo Coppola, Pietro Liò

1Independent Researcher · Department of Computer Science and Technology, University of Cambridge

Graph Neural Networks Graph Representation Learning Message Passing Neural Networks Sheaf Neural Networks Sheaf Theory

Aug 2, 2026eess.SP

Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling

Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines.

Gabriele D'Acunto, Leonardo Di Nino, Paolo Di Lorenzo +1

Department of Information Engineering, Electronics, and Telecommunications, Sapienza University of Rome, 00184 Rome, Italy · National Inter-University Consortium for Telecommunications (CNIT), Parma, Italy

Graph Signal Processing Sheaf Theory Spectral Methods

Jul 28, 2026cs.LG

Learned, Relied Upon, or Necessary? Separating Checkpoint Dependence from Task-Level Value in Sheaf GNNs

Learned restriction maps in sheaf graph neural networks are often treated as proof that the model has discovered useful edge geometry. That conclusion does not follow from parameter movement or from a post-hoc ablation: both can show how one checkpoint is organized while leaving open whether learned transport still helps after the rest of the model adapts. We separate these claims with two estimands. Checkpoint reliance intervenes on the maps of a fixed predictor; protocol-relative replacement retrains matched families that remove map capacity, edge variation, or persistent edge assignment. A task-null theorem shows why the claims can diverge: labels identify only the transported classifier directions, leaving

d^2-d

invisible degrees of freedom in every full

d\times d

map. An exact frame model then gives the boundary at which reliance becomes unreplaced task value. Label-only training realizes the predicted separation, while audits of public NSD, DNSD, and Directed Sheaf Neural Network (DSNN) implementations recover both replaceable and unreplaced transport regimes on real graphs. All five DNSD benchmarks exhibit fixed-checkpoint reliance. After retraining, assignment-breaking or shared-map controls recover Full performance on four; Roman-Empire retains a

.0675

advantage over continually resampled assignment and a

.0391

advantage over a parameter-matched shared map across ten official splits. Thus, a learned map can govern a fitted computation without constituting indispensable edge geometry. Claims of learned transport should pair checkpoint interventions with matched retraining.

Yi Liu

School of Astronomy and Space Science, University of Science and Technology of China

Sheaf Neural Networks Sheaf Theory GNN Explainability

Jul 21, 2026cs.LG

Do Sheaf Neural Networks Use Holonomy? A Measure--Intervene--Control Study

Geometric architectures are often justified by internal mechanisms such as rotations, yet task performance alone cannot show whether those mechanisms drive predictions. Using sheaf neural networks (SNNs) as a testbed, we introduce the first basis-independent measurement of trained triangle-loop products, separating rotation, stalk-space area, and orientation. In a custom high-homophily GraphUniverse regime, Neural Sheaf Propagation (NSP) increases the triangle-weighted mean two-dimensional SO(2) loop rotation from 0.010 to 0.388 radians for triangle counting, while the community-detection comparison ends at 0.029 radians. Across the training-set-size experiment, replacing all learned SO(2) transports with identities sharply increases test error, establishing post-training sensitivity to the complete learned connection. However, a graph-summary ridge predictor is more accurate, diagonal maps also improve, and fixed-degree graphs develop increasing rotation without outperforming the training-mean predictor. This measure-intervene-control study separates geometric change, connection sensitivity, and evidence for triangle-specific computation.

Ankit Grover, Rémi Bourgerie

KTH Royal Institute of Technology, Stockholm, Sweden

Neural Network Interpretability Sheaf Neural Networks Sheaf Theory

Jun 18, 2026cs.LG

Hierarchical Pooling for Sheaf Neural Networks

Sheaf Neural Networks (SNNs) generalize Graph Neural Networks (GNNs) by replacing scalar node signals with stalk-valued signals and by using restriction maps to measure compatibility across edges. Unlike standard graph diffusion, which encourages neighboring node features to become similar, sheaf diffusion promotes consistency through the restriction maps and can therefore model more general relationships between neighboring nodes. However, existing sheaf neural architectures mainly operate at a fixed graph resolution and do not provide a principled pooling mechanism for building hierarchical representations. In this paper, we introduce Hierarchical Sheaf Pool (HiSP), a sheaf-aware pooling framework based on local spectral coarsening. Given a partition of the graph, HiSP constructs each coarse stalk by projecting fine stalk-valued features onto the low-frequency eigenmodes of the cluster-internal sheaf Laplacian. These local modes define a cochain-level prolongation map, which allows the fine sheaf energy to be represented on the coarse space through a Galerkin operator. We further analyze the approximation induced by coarsening by separating truncation loss, due to discarded local modes, from realization loss, due to representing the projected operator as a coarse sheaf. Finally, we implement HiSP as a GNN pooling layer compatible with SNNs and provide a PyG implementation supporting batching, lifted sheaf Laplacians, and hierarchical architectures.

Dionisia Naddeo, Carlo Abate, Pietro Liò +2

UiT The Arctic University of Norway · UiT The Arctic University of Norway NORCE Norwegian Research Centre AS

Graph Neural Networks Graph Coarsening Graph Pooling Hierarchical Representation Learning Sheaf Neural Networks Sheaf Theory

Jun 10, 2026cs.CV

SheafStain: Sheaf-Theoretic Schrödinger Bridge for Spatially and Biologically Coherent Virtual Staining

Current virtual staining approaches offer the potential for time- and cost-efficient biomarker quantification in cancer diagnostics and prognostics. However, patch-wise inference for gigapixel whole-slide images (WSIs) fails to maintain spatial continuity, yielding artifacts that cause catastrophic mismatches with ground-truth images. Although pathology Vision Foundation Models (VFMs) offer rich representations, their self-attention causes varying global contexts to produce inconsistent embeddings for the same physical region. We formalize and validate this ``context contamination'' as a sheaf-theoretic problem where these embeddings form a presheaf whose sections disagree on overlaps, so no global section restricts to them. To address this, we propose SheafStain, a new approach that reinterprets VFM features as sheaf-like sections for spatially and biologically coherent virtual staining. Specifically, SheafStain integrates class and patch tokens into a Schrödinger Bridge framework as sheaf-like sections. While the class token anchors biological consistency, patch tokens form a per-position spatial map. An encoder co-pretrained on Hematoxylin & Eosin (H&E) and Immunohistochemistry (IHC) yields cross-stain sections, so a single VFM feature space supervises both input conditioning and output stain alignment. Departing from prior work that evaluates on isolated

256 \times 256

patches and either random-crops or resizes the

1024 \times 1024

ground truth, we translate at

256 \times 256

and evaluate on the stitched

1024 \times 1024

outputs across HER2, ER, PR, and Ki-67. SheafStain demonstrates promising results against six prior methods while mitigating patch-boundary stitching artifacts. Code is available at https://github.com/deepnoid-ai/SheafStain.

Hyeongyeol Lim, Hongjun Yoon, Eunjin Jang +3

Department of Medical Informatics, College of Medicine, Korea University · DEEPNOID Inc.

Computational Pathology Vision Foundation Models Virtual Staining Sheaf Theory Whole-Slide Image Analysis

Jun 2, 2026cs.DS

Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry

We present an algorithmic framework for incremental maintenance of first sheaf cohomology

H^1(X; \mathcal{F})

on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of

H^1

via factorization of the coboundary matrix requires

O(n^3)

time; when the complex evolves with a stream of

m

edits, full recomputation after each edit costs

O(mn^3)

. Under a bounded local geometry assumption -- bounded cell size

v_{\max}

, bounded stalk dimension

d

, and bounded nerve degree

D

-- each edit (vertex insertion, edge insertion, restriction map update) affects only a bounded set of local coboundary blocks. The algorithm therefore processes lazy streaming edits in

O(1)

time with respect to the total complex size

n

(with cost polynomial in the local geometry parameters

v_{\max}

,

d

, and

D

, which are treated as constants independent of

n

), deferring local eigensolves and Mayer-Vietoris global assembly to synchronization points (Flush). At synchronization, the maintained state agrees with the corresponding batch assembly of the partitioned sheaf model; we observe zero measured drift in all batch-verified runs (through

V = 10^6

). We also give an amortized

O(|E|)

streaming construction for the cellular decomposition and discuss an adversarial algebraic-RAM barrier arguing that unpartitioned non-trivial sheaves (

d \geq 2

, non-identity restriction maps) do not admit the same locality. Experiments on Barabasi-Albert graphs with up to

5 \times 10^6

vertices and

1.7 \times 10^7

streaming edits show 35

μ

s median lazy per-edit update latency (excluding flush); query time (global assembly at synchronization) is

O(n)

per flush in the implemented full-traversal path. Exact synchronization costs are reported separately.

Jason L. Volk

Invariant Research

Sheaf Theory Streaming Algorithms

Jun 1, 2026cs.GT

A Sheaf Framework for Strategic Multi-Agent Systems: From Consensus to Nash Equilibria

The coordination of heterogeneous autonomous agents in dynamic, adversarial environments requires simultaneous satisfaction of geometric constraints, logical consistency, temporal reasoning, and strategic optimization. Existing sheaf- and topos-theoretic frameworks provide powerful tools for geometric consensus, knowledge alignment, and causal planning, but lack explicit models for value, reward, and strategic choice. This report presents a unified categorical framework that integrates event calculus, SCEL-like ensemble formation, and game-theoretic reward structures into a single Grothendieck topos of time-space histories. We introduce the notion of a \emph{game sheaf} whose stalks contain utility functions and policy distributions, and restriction maps encode both parallel transport and best-response dynamics. We prove that Nash equilibria correspond to global sections of a derived best-response correspondence sheaf, while cohomological obstructions classify failures of strategic consistency. A detailed case study of an immunological ``bastion defense'' scenario -- heterogeneous agents forming attack/defense ensembles under resource constraints -- demonstrates the framework's expressiveness. This synthesis provides a rigorous foundation for verifiable, autonomic, and economically rational multi-agent systems.

Manuel Hernández, Eduardo Sánchez-Soto

Nash Equilibrium Multi-Agent Coordination Game Theory Sheaf Theory Multi-Agent Systems

May 20, 2026cs.LG

Gaussian Sheaf Neural Networks

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.

André Ribeiro, Ana Luiza Tenório, Tiago da Silva +1

Getulio Vargas Foundation · MBZUAI

Graph Neural Networks Sheaf Neural Networks Sheaf Theory

May 14, 2026cs.LG

TopoPrimer: The Missing Topological Context in Forecasting Models

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.

Zara Zetlin, Kayhan Moharreri, Maria Safi

Apple

Time Series Forecasting Sheaf Theory Time-Series Foundation Models

May 13, 2026cs.AI

Sheaf-Theoretic Transport and Obstruction for Detecting Scientific Theory Shift in AI Agents

Scientific theory shift in AI agents requires more than fitting equations to data. An artificial scientific agent must detect whether an existing representational framework remains transportable into a new regime, or whether its language has become locally-to-globally obstructed and must be extended. This paper develops a finite sheaf-theoretic framework for detecting theory-shift candidates through transport and obstruction. Contexts are organized as a local-to-global structure in which source, overlap, target, and validation charts are fitted, restricted, and tested for gluing. Obstruction measures failure of coherence through residual fit, overlap incompatibility, constraint violation, limiting-relation failure, and representational cost. We evaluate the framework on a controlled transition-card benchmark designed to separate deformation within a source language from extension of that language. The main result is direct obstruction ranking: the intended deformation or extension is usually the lowest-obstruction candidate, and transition type is separated in the benchmark. A constellation kernel over the same signatures is included only as a secondary representational-similarity probe. The aim is not to reconstruct historical paradigm shifts or solve open-ended autonomous theory invention, but to isolate a finite diagnostic subproblem for AI agents: detecting when representational transport fails and extension becomes the coherent next move.

David N. Olivieri, Roque J. Hernández

Universidade de Vigo, Department of Computer Science (LSI), Spain

AI Agents for Scientific Discovery Sheaf Theory Semantic Change Detection

May 11, 2026eess.SY

Multi-Agent System Identification with Nonlinear Sheaf Diffusion

Local interaction laws governing multi-agent systems can be difficult to recover from trajectory data, even when the dynamics are observed faithfully. In systems governed by a nonlinear sheaf Laplacian -- a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels -- the coordination law is encoded by edge potential functions whose gradients produce the inter-agent forces. Because trajectory observations record node-state evolution, they expose only the aggregate effect of the edge forces at each node: distinct interaction laws that agree at the node level are indistinguishable from trajectory data alone. We show that the fundamental obstruction to recovery is topological, measured by sheaf cohomology, and that unique recovery from an unconstrained function class is possible if and only if this cohomology vanishes. When the obstruction is nontrivial, we show that recovery within a finite-dimensional parameterized class is possible precisely when a data-dependent information matrix is positive definite. Experiments validate the theory and illustrate that accurate trajectory reproduction need not certify recovery of the underlying interaction law.

Nivar Anwer, Hans Riess, Matthew Hale

Department of Computer Science, Georgia Tech · Department of Electrical & Computer Engineering, Georgia Tech

Parameter Identifiability Nonlinear System Identification Sheaf Theory Multi-Agent Systems

May 11, 2026cs.LG

Oversmoothing as Representation Degeneracy in Neural Sheaf Diffusion

Neural Sheaf Diffusion (NSD) generalizes diffusion-based Graph Neural Networks by replacing scalar graph Laplacians with sheaf Laplacians whose learned restriction maps define a task-adapted geometry. While the diffusion limit of NSD is known to be the space of global sections, the representation-theoretic structure of this harmonic space remains largely implicit. We develop a quiver-theoretic interpretation of NSD by identifying cellular sheaves on graphs with representations of the associated incidence quiver. Under this correspondence, learned sheaf geometries become points in a finite-dimensional representation space. We show that direct-sum decompositions of the underlying incidence-quiver representation induce decompositions of the harmonic space reached in the diffusion limit. This gives an algebraic interpretation of oversmoothing as representation degeneration: learned sheaves may collapse toward low-complexity summands whose global sections fail to preserve discriminative information. Building on this viewpoint, we connect sheaf diffusion to stability and moment-map principles from Geometric Invariant Theory. We introduce moment-map-inspired regularizers that bias restriction maps toward balanced representation geometries, and identify a structural obstruction in equal-stalk architectures: when

d_v = d_e

, admissibility for learnable stability parameters forces the trivial all-object summand onto a stability wall. Non-uniform stalk dimensions remove this obstruction, making adaptive stability meaningful. Experiments on heterophilic benchmarks are consistent with this mechanism: breaking stalk symmetry can reduce variance or improve validation behavior, and adaptive stability becomes more effective in selected rectangular settings. Overall, our framework reframes oversmoothing as a degeneration phenomenon in the representation geometry underlying learned sheaf diffusion.

Arif Dönmez, Axel Mosig, Ellen Fritsche +1

IUF – Leibniz Research Institute for Environmental Medicine, Düsseldorf, Germany · DNTOX GmbH, Düsseldorf, Germany · Bioinformatics Group, Ruhr University Bochum, Bochum, Germany +1

Graph Neural Networks Sheaf Neural Networks Sheaf Theory Graph Diffusion

May 3, 2026cs.AI

Sheaf-Theoretic Planning: A Categorical Foundation for Resilient Multi-Agent Autonomous Systems

The challenge of engineering autonomous agents capable of navigating the stochastic and adversarial nature of the physical world has historically resided at the intersection of symbolic logic and control theory. Traditional multi-agent system (MAS) frameworks have relied heavily on monolithic logical models -- primarily variations of the event calculus and situation calculus -- to represent action, change, and temporal persistence. While these classical systems provide robust solutions to the frame problem through mechanisms like circumscription and successor state axioms, they are inherently limited by a closed-world assumption that fails in the face of unobserved agent interventions, plan interruptions, and divergent belief-reality states. The paradigm of Sheaf-Theoretic Planning (STP) emerges as a transformative alternative, grounding the problem of multi-agent coordination under the mathematical structures of topos theory and sheaf semantics. This report provides an exhaustive analysis, justification, and extension of the STP framework, exploring its categorical foundations, implementation feasibility, and role in the future of resilient autonomous systems.

Manuel Hernández, Eduardo Sánchez-Soto

Universidad Tecnol´ogica de la Mixteca

Multi-Agent Coordination Sheaf Theory Multi-Agent Planning

Apr 22, 2026cs.LG

Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-specific transformations, but existing formulations remain confined to vector spaces and therefore cannot propagate matrix-valued features. We address both challenges by developing the first sheaf neural network operates natively on the SPD manifold. Our key insight is that the SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space. Theoretically, we prove that SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations. Empirically, our sheaf convolution transforms effectively rank-1 directional inputs into full-rank matrices encoding local geometric structure. Our dual-stream architecture achieves SOTA on 6/7 MoleculeNet benchmarks, with the sheaf framework providing consistent depth robustness.

Yuhan Peng, Junwen Dong, Yuzhi Zeng +6

School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore · Chern Institute of Mathematics, Nankai University, Tianjin, China · Department of Mathematics, Faculty of Science, National University of Singapore, Singapore +3

Graph Neural Networks Geometric Representation Learning Manifold Learning Sheaf Neural Networks Sheaf Theory Molecular Representation Learning