RecBundle: A Next-Generation Geometric Paradigm for Explainable Recommender Systems

RecBundle: A Next-Generation Geometric Paradigm for Explainable Recommender Systems Hui Wang ∗ wanghui4042@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences Beijing, China Tianzhu Hu ∗ hutianzhu@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences Beijing, China Mingming Li † limingming@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences Beijing, China Xi Zhou zhouxi@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences Beijing, China Chun Gan † ganchun@jd.com JD.com Beijing, China Jiao Dai daijiao@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences Beijing, China Jizhong Han hanjizhong@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences Beijing, China Songlin Hu husonglin@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences School of Cyberspace Security, University of Chinese Academy of Sciences Beijing, China Tao Guo guotao@iie.ac.cn Institute of Information Engineering, Chinese Academy of Sciences Beijing, China Abstract Recommender systems are inherently dynamic feedback loops where prolonged local interactions accumulate into macroscopic structural degradation such as information cocoons. Existing rep- resentation learning paradigms are universally constrained by the assumption of a single flat space, forcing topologically grounded user associations and semantically driven historical interactions to be fitted within the same vector space. This excessive coupling of heterogeneous information renders it impossible for researchers to mechanistically distinguish and identify the sources of systemic bias. To overcome this theoretical bottleneck, we introduce Fiber Bundle from modern differential geometry and propose a novel geometric analysis paradigm for recommender systems. This the- ory naturally decouples the system space into two hierarchical layers: the base manifold formed by user interaction networks, and the fibers attached to individual user nodes that carry their dy- namic preferences. Building upon this, we construct RecBundle, a framework oriented toward next-generation recommender systems ∗ Both authors contributed equally to this research. † Corresponding authors. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. Conference’17, Washington, DC, USA © 2026 Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 978-x-x-x-x/Y/M https://doi.org/10.1145/n.n that formalizes user collaboration as geometric connection and parallel transport on the base manifold, while mapping content evo- lution to holonomy transformations on fibers. From this foundation, we identify future application directions encompassing quantita- tive mechanisms for information cocoons and evolutionary bias, geometric meta-theory for adaptive recommendation, and novel inference architectures integrating large language models (LLMs). Empirical analysis on real-world MovieLens and Amazon Beauty datasets validates the effectiveness of this geometric framework. CCS Concepts • Information systems→ Recommender systems. Keywords Fiber Bundle, Recommender systems, Explainable ACM Reference Format: Hui Wang, Tianzhu Hu, Mingming Li, Xi Zhou, Chun Gan, Jiao Dai, Jizhong Han, Songlin Hu, and Tao Guo. 2026. RecBundle: A Next-Generation Geo- metric Paradigm for Explainable Recommender Systems. In . ACM, New York, NY, USA, 6 pages. https://doi.org/10.1145/n.n 1 Introduction With the deep integration of mobile internet and artificial intelli- gence, recommender systems have evolved into critical nexuses connecting massive data with individual users. Yet these systems constitute dynamic feedback loops where user behaviors, content attributes, and algorithmic strategies interact and co-evolve over time [30]. The accumulation of micro-level local decisions through arXiv:2603.16088v1 [cs.IR] 17 Mar 2026 Conference’17, July 2017, Washington, DC, USAHui Wang, Tianzhu Hu, Mingming Li, Xi Zhou, Chun Gan, Jiao Dai, Jizhong Han, Songlin Hu, and Tao Guo these loops ultimately confronts systems with deep-rooted chal- lenges including interpretability deficits and structural fragility, which macroscopically manifest as information cocoons. Although recent advances in graph neural networks [18], multi-objective op- timization, causal inference, and generative models [29,34,36,40] have achieved remarkable progress, these methods predominantly focus on static preference fitting and observational statistical cor- rections, lacking a rigorous mathematical characterization of how information propagates and evolves withi the n user-item space. A fundamental limitation underlies existing paradigms. Whether employing Euclidean spaces or recent hyperbolic and Riemannian geometries, they invariably force user preferences and item features into a single global metric space. This inherently couples two hetero- geneous mechanisms that drive information flow in recommender systems. The first is population-level collaboration, namely horizon- tal information diffusion through user connection structures. The second is individual-level evolution, or vertical accumulation of dy- namic preferences via semantic continuity of historical interactions. By forcing these heterogeneous signals into identical representa- tion spaces for joint optimization, mainstream approaches make it theoretically impossible to pinpoint the origin of systemic biases when they emerge. To resolve this, we turn to Fiber Bundle theory [3,41]. Widely adopted in gauge theory [17] and computer vision [5,23], this theory has proven effective in describing how complex systems assemble local product structures into intricate global topologies [25–27]. It models the system as a composite object. A base manifold formed by the discrete user interaction graph encodes topological distances and collective collaborative relationships. Fibers attached to individual user nodes carry continuous preference evolution. This orthogonal decoupling separates who resembles whom from how a specific user evolves, enabling separate modeling of collective structure and individual dynamics. Building on this insight, we propose RecBundle, a geometric analysis framework for next-generation explainable recommender systems. Using the language of Fiber Bundle, this framework mathe- matically reconstructs collaborative and evolutionary mechanisms. It formalizes inter-user information collaboration as geometric con- nections and parallel transport on the base manifold. It maps long- term preference evolution to holonomy transformations on fiber spaces. This theoretical foundation provides unified mathematical grounding for the following contributions: •A quantifiable framework for information evolution. It translates phenomena like information cocoons into geo- metric anomalies: local curvature distortion and holonomy- induced dimensional contraction, providing rigorous tools for analyzing information degradation. • A geometric meta-theory for adaptive recommenda- tion. It reconceptualizes personalization as geometric con- struction, where algorithms dynamically match manifold structures to each user’s cognitive state. •A novel reasoning architecture integrating Large Lan- guage Models. It maps fast-and-slow thinking onto Fiber Bundle geometry, using cross-node retrieval for rapid align- ment and fiber constraints to ensure semantic consistency in chain-of-thought reasoning. 2 Preliminaries This section briefly reviews the Fiber Bundle theory, providing a unified mathematical language for decoupling topological struc- tures and dynamic semantics in recommender systems. 2.1 Topological Structure Mathematically, a fiber bundle [22] (퐸,퐵,퐹 ,휋 ) decouples a complex state space into a base manifold퐵that serves as the topological skeleton whose points represent distinct entities, with attached fibers퐹that function as independent state spaces characterizing internal dynamics. The projection map휋:퐸 → 퐵anchors high- dimensional states in the total space퐸to specific entities on퐵, thereby orthogonally decoupling an entity’s physical topology from its internal semantics, as shown in Figure 1. Specifically, when the fiber itself is a Lie group퐺, this structure constitutes a principal bundle [17], which is rigorously character- ized by a free and smooth right group action on the total space, a surjective projection mapping, and equivariant local trivializations that ensure global structural consistency. 1 G π(p) g π E e p pg B Figure 1: Simple Diagram of a Principal Fiber Bundle. 2.2 Evolution Operators Since fibers at different manifold points are mutually independent vector spaces, direct cross-node algebraic operations are geometri- cally invalid. To enable cross-node information passing, a connec- tion휔must be introduced, which defines how a fiber state at one point can be parallel transported along a path on the base manifold to the fiber of an adjacent point. However, when the base manifold is non-flat, i.e., possessing non-zero curvatureΩ, this parallel trans- port exhibits strong path dependence. If a state vector is parallel transported along a closed loop훾on the base manifold back to its starting point, an irreversible linear transformation typically occurs, known as a holonomy transformation퐻표푙(훾). This geometric pro- jection and dimensionality reduction effect induced by closed-loop paths provides a precise mathematical tool for quantifying state contraction in the long-term evolution of complex systems. 3RecBundle: A Geometric Analysis Framework To introduce the mathematical constructs of Fiber Bundle into the practical modeling of recommender systems, this chapter proposes the RecBundle (Recommendation on Fiber Bundle) analysis frame- work. This framework aims to establish a formal correspondence RecBundle: A Next-Generation Geometric Paradigm for Explainable Recommender SystemsConference’17, July 2017, Washington, DC, USA Algorithm Content User User Behavior & feedbacks Recommendation List Content Engagement User Interaction Filtering /Matching Content Features Information Cycle (a) Information Flow(Macro) Base Manifold B （User Space） Fiber F （Preferences） ᵞ Connection ω Total Space E (b) Fiber Bundle Formalization Figure 2: Paradigm Shift in Recommender Systems Modeling. between the information passing mechanisms of recommender systems and manifold geometric operators. From this perspective, we provide theoretical support for quantifying macro-level biases during the system’s dynamic evolution. 3.1 Geometric Mapping of the Dual Collaborative Mechanisms In the RecBundle framework, we bridge the macroscopic informa- tion cycle of recommender systems with rigorous mathematical constructs, as illustrated in Figure 2a. Specifically, we formalize the correspondence between these core system elements and the geometric objects of Fiber Bundle, as shown in Figure 2b. As sum- marized in Table 1 and 2, this geometric mapping mathematically decouples the recommendation process. The base manifold captures the collaborative user topology, while the attached fibers model dynamic preference updates. This formulation not only unifies mainstream recommendation paradigms as special geometric cases, but also introduces curvature and holonomy as concrete metrics to quantify system biases and structural vulnerabilities. 3.2 Parallel Transport Horizontal collaboration in recommender systems aims to leverage similar users to supplement target user information, e.g., neigh- borhood aggregation in GNNs. From the fiber bundle perspective, since the preference features of users푣and푢, denoted as푓 푣 and푓 푢 respectively, reside in mutually independent fiber spaces퐹 푣 and퐹 푢 , performing direct addition or inner product operations on them in Euclidean space is generally geometrically ill-posed. This rep- resents a primary cause of the coupling of heterogeneous signals. To achieve rational information collaboration, recommendation algorithms essentially learn a discrete geometric connection휔im- plicitly. The aggregation process is equivalent to utilizing a parallel transport operator푃 푣→푢 to transport the preference state퐹 푣 of neighbor푣along the base manifold and project it into the tangent space of the target user푢, as illustrated in Figure 3a. 푃 푣→푢 (푓 푣 ) ≈ 훼 푢푣 W푓 푣 .(1) In this discretized representation, the weight matrix W corre- sponds to the directional component of the connection, responsible for transforming the feature bases, while the attention coefficient 훼 푢푣 corresponds to the intensity component of the connection. When local interactions are sparse or user heterogeneity is strong, the local connection curvatureΩof the base manifold is typically high. In such cases, enforcing cross-region parallel trans- port to fit global objectives easily generates significant local spatial distortions. This perspective provides a geometric explanation for why GNNs are prone to over-smoothing [4] on tail users and offers a reference for designing robust aggregation mechanisms. 3.3 Holonomy Transformations Besides horizontal collaboration, the other core mechanism of rec- ommender systems is the dynamic closed loop involving interaction- recommendation-feedback. In RecBundle, we view a user’s sequen- tial interaction history as a path훾on the base manifold퐵. As a user continuously receives recommendations and provides feedback, their initial preference state푓 푖푛푖푡 evolves within the fiber space퐹 푢 . After experiencing one or more feedback loops훾, influenced by the cumulative effect of algorithmic connections, the state vector un- dergoes a corresponding linear transformation:푓 푒푛푑 = 퐻표푙(훾)· 푓 푖푛푖푡 . This transformation matrix induced by the closed-loop path is known as the Holonomy, as shown in Figure 3b. From geometric insight, in an ideal setting,퐻표푙(훾)should ap- proximate a volume-preserving orthogonal transformation, sup- porting natural interest migration without dimensional loss. How- ever, under existing algorithms, the holonomy operator manifests as a contraction mapping: spectral decomposition reveals that non- dominant interest components are progressively suppressed during closed-loop evolution. This volume contraction provides a funda- mental mathematical characterization of information dynamics in recommender systems. 4 Evolution Blueprint Building on the RecBundle theoretical framework established ear- lier, this section demonstrates how this geometric paradigm can address fundamental challenges in recommender systems. We out- line its application potential from three perspectives: explainable quantification of information evolution, adaptive recommendation mechanisms, and a novel inference architecture that integrates large language models. 4.1An Explainable Framework for Information Evolution Most existing research on recommender system explainability re- lies on post-hoc attribution or static weight analysis. These ap- proaches fail to capture the underlying mechanisms of continuous information evolution [1,2,14,19]. RecBundle introduces a process- oriented explainability paradigm that unifies three progressively severe phenomena of information degradation: recommendation bias [10,11,15], filter bubbles [28,32,33], and rumor propagation [12,35]. The framework integrates these phenomena into a single mathematical formulation based on geometric invariants, specif- ically curvature and holonomy. It maps these phenomena onto directional shifts and volumetric contraction in the representation manifold, enabling white-box analysis of system dynamics. Consider information cocoons as an example. Geometrically, information cocoons are characterized as volumetric shrinkage in the feature space. This quantification begins at the local interac- tion level. When the model aggregates heterogeneous features, it Conference’17, July 2017, Washington, DC, USAHui Wang, Tianzhu Hu, Mingming Li, Xi Zhou, Chun Gan, Jiao Dai, Jizhong Han, Songlin Hu, and Tao Guo Table 1: Correspondence between Recommender System Concepts and Fiber Bundle Theory RS ConceptFiber Bundle ObjectSymbolPractical Meaning & Function User Set & TopologyBase Space퐵(푈)Network topology encoding user cognitive distances. Preferences SpaceFiber퐹 푢 Independent semantic spaces for dynamic user interests. Sequential InteractionsPath on Manifold훾(푡)Historical interaction trajectories or feedback loops. Feature AlignmentConnection휔Geometric rules for cross-node feature alignment. CollaborativeParallel Transport푃 푢→푣 Projection operators for neighbor feature aggregation. Local BiasConnection CurvatureΩSpatial distortions during aggregation. Long-term EvolutionHolonomy Transformation퐻표푙(훾)Closed-loop evolving transformations. Parallel Transport Intermediate users Initial preference state Parallel Transport Predicted preference state ... ... Base Manifold B （User Space） (a) Collaborative aggregation (parallel transport) Parallel Transport Intermediate users Initial perference state Parallel Transport ... ... Base Manifold B （User Space） ... ... Closed Loop Holonomy Group After Hol-transfer Semantic Distortion (b) Preference evolution (holonomy transformation) Figure 3: Core geometric operators in RecBundle. (a) Parallel transport aligns heterogeneous neighbor features across the base manifold. (b) Closed feedback loops induce holonomy transformations. Table 2: Geometric Interpretations of Mainstream Recommendation Paradigms Algorithm ParadigmBase Manifold AssumptionCurvature FeaturesInterpretations & Limitations from a Geometric Perspective Matrix FactorizationEuclidean flat spaceZero curvatureStatic space; ignores path dependence and state dynamics. Graph Neural NetworksDiscrete graph manifoldImplicit linear Linearized transport; over-smooths in high-curvature regions. Large Language ModelsDiscrete token sequenceAttention weightsUnconstrained paths; prone to semantic hallucinations. RecBundle (Ours)Stratified manifoldExplicit holonomyDecouples topology & semantics; quantifies evolutionary bias. introduces geometric distortion captured by the connection curva- tureΩ. For discrete graphs, we approximateΩusing local feature alignment residuals: ˆ Ω 푢 = 1 N(푢) ∑︁ 푣∈N(푢) 훼 푢푣 ·∥f u − Wf 푣 ∥ 2 .(2) This computation relies solely on the model’s native latent states, soΩis strictly differentiable. A highΩvalue indicates approxi- mation errors when modeling discrete preference transitions. As local interactions accumulate along feedback paths훾, long-term evolution is governed by the holonomy matrix H 훾 , defined as the ordered product of update Jacobians J 푡 . We evaluate the spectral radius휌= 푚푎푥|휆 푘 |to measure maximum semantic contraction. We then introduce the Geometric Bias Index (GBI) to capture global volume loss: 퐺퐵퐼= 1− 1 푑 퐿 ∑︁ 푘=1 |휆 푘 (H 훾 )|,(3) where휆 푘 represents the k-th eigenvalue of the holonomy matrix, and푑is the feature dimension of the fiber space. As GBI approaches 1, it indicates that numerous orthogonal feature dimensions have been compressed during evolution, leading to severe degradation of the semantic space. The proposed metrics enable quantitative evaluation in prac- tical recommendation tasks. As shown in Figure 4, GBI exhibits a strong negative correlation with Shannon entropy, confirming the consistency between our geometric framework and traditional diversity metrics. Furthermore, Table 3 provides architectural in- sights at two levels. At the data level, sparsity serves as a primary catalyst for structural contraction , such as Beauty dataset, leading to sharp increases in bothΩand휌. At the model level, under sparse conditions, architectures with bidirectional dependencies such as BERT4Rec demonstrate significantly higher curvatureΩcompared to unidirectional models such as SASRec. This reveals that complex attention mechanisms are more susceptible to alignment errors RecBundle: A Next-Generation Geometric Paradigm for Explainable Recommender SystemsConference’17, July 2017, Washington, DC, USA 1.61.82.02.22.42.6 Shannon Entropy (Diversity ) 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 Holonomy Radius (Contraction ) Geometric Dynamics: Entropy vs. Radius Selected Samples Trend Line (r =0.802) Q1 (Cocoon)Q2Q3Q4Q5 (Explorer) User Diversity Quantiles 0.76 0.77 0.78 0.79 0.80 0.81 0.82 Holonomy Radius Distribution of by Entropy RecBundle: Geometry of Information Cocoons (Validated on ML-1M) Figure 4: The correlation between Shannon Entropy and spec- tral radius. when observation samples are limited, making them more prone to information cocoons. Table 3: Geometric features across real world datasets Models ML-1MBeauty Ω휌GBIΩ휌GBI SASRec0.25140.14320.11300.45340.32780.0780 BERT4Rec0.37600.12250.08700.97080.37870.0911 This quantification logic extends naturally from information cocoons to rumor propagation. When local geometric deviations accumulate across multiple propagation nodes, they induce sub- stantial directional shifts in feature vectors. This continuous ampli- fication mechanism explains how misinformation exploits network topology to bypass standard semantic constraints. To address these issues, future optimization can directly regular- ize specific parameters to maintain stable evolution. First, curvature regularization can optimize attention weights and projection matri- ces:L 푡표푡푎푙 =L 푡푎푠푘 + 휆· 1 |푈| Í 푢∈푈 ˆ Ω 푢 , suppressing high-curvature regions at their source. Second, holonomy constraints can regular- ize sequence encoders by imposing volume preservation conditions (| det(J 푡 )| ≈1) or spectral penalties on update Jacobians, forcing temporal parameters to preserve the full dimensionality. 4.2 Metatheory for Adaptive Recommendation The Fiber Bundle framework provides a unifying geometric lan- guage for adaptive learning systems, revealing that a broad class of meta-learning algorithms [6,9,16,20,21,24,31] can be understood as special cases of this structure. In this view, the base manifold encodes the space of meta-parameters, while each fiber attached to a base point contains task-specific parameters reachable via adap- tation. The inner-loop update process is geometrically interpreted as parallel transport along the fiber, with different meta-learning methods corresponding to distinct choices of the connection that governs this transport. For instance, standard MAML with full dif- ferentiation implicitly learns a connection capturing second-order curvature [9], whereas first-order approximations assume a trivial connection, and methods like iMAML impose a Levi-Civita-like connection through explicit regularization [31]. Beyond these longitudinal relationships between meta and task parameters, many real-world scenarios exhibit transverse relation- ships among tasks themselves, such as in federated learning or cross-domain transfer. This motivates extending the fiber bundle to a principal bundle where a structure group acts on fibers, encoding task correlations; methods like HSML that cluster tasks effectively learn such group actions [38], and the resulting holonomy quan- tifies task interdependence via curvature. Future work will focus on learning bundle geometry end-to-end from data, introducing curvature regularization to enhance knowledge transfer and gener- alization across tasks, and extending the framework to multimodal scenarios, thereby laying a rigorous theoretical foundation for de- signing next-generation adaptive algorithms. 4.3 A Novel Inference Architecture with LLMs As large language models reshape recommender systems, recom- mendation tasks are evolving from pattern matching toward deep semantic reasoning. State-of-the-art LLMs increasingly explore so- phisticated test-time computation and slow thinking mechanisms [7,8,13,37,39]. However, without explicit structural constraints, LLMs’ processing long interaction sequences often deviates from users’ interest boundaries, producing factual errors or logical dis- continuities known as semantic hallucinations. The hierarchical geometry of RecBundle provides a natural frame- work for integrating LLMs reasoning. Cross-node routing on the base manifold퐵corresponds to System 1 fast retrieval, using the user-item topological network to bound information extraction. Transitions along a user fiber퐹 푢 correspond to System 2 deep chain-of-thought reasoning. Each autoregressive generation step can be formalized as local parallel transport along the fiber. Trans- lating geometric invariants into decoding constraints, such as in- corporating curvature as regularization in attention mechanisms or constraining transformation matrices to maintain local smooth- ness, mechanistically guides the reasoning trajectory. This ensures semantic consistency across multi-step iterations. This paradigm points toward more trustworthy generative recommendations. 5 Conclusion Grounded in the physical and mathematical boundaries of informa- tion flow in recommender systems, this work introduced RecBun- dle, the first geometric paradigm to provide a unified mathematical language for understanding information evolution in complex envi- ronments. Building on this geometric foundation, we identify three future directions. First, geometric priors can be integrated as differ- entiable constraints during training to improve generalization for long-tail users and cold-start items. Second, constructing smooth mappings between heterogeneous manifolds enables knowledge transfer across domains and modalities. Third, structure-aware agents combining local perception with reinforcement learning can dynamically adjust exploration-exploitation trade-offs. We hope RecBundle will inspire a fundamental re-examination of the dynamic mechanisms underlying recommender systems, fostering next-generation paradigms that unite theoretical inter- pretability with long-term robustness. Conference’17, July 2017, Washington, DC, USAHui Wang, Tianzhu Hu, Mingming Li, Xi Zhou, Chun Gan, Jiao Dai, Jizhong Han, Songlin Hu, and Tao Guo References [1]Qifeng Bai, Nankai Lin, Meiyu Zeng, Guanqiu Qin, Dong Zhou, and Aimin Yang. 2025. Ensuring accuracy and fairness: a de-biasing framework for sequential recommendation. User Modeling and User-Adapted Interaction 35, 2 (2025), 9. [2]Tian Bian, Xi Xiao, Tingyang Xu, Peilin Zhao, Wenbing Huang, Yu Rong, and Junzhou Huang. 2020. Rumor Detection on Social Media with Bi-Directional Graph Convolutional Networks. 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