ManifoldMind: Probabilistic Hyperbolic Recommender

• ManifoldMind is a probabilistic geometric recommender system that embeds users, items, and tags into hyperbolic space using adaptive-curvature probabilistic spheres.
• It employs a curvature-aware semantic kernel and beam search for efficient multi-hop inference to traverse semantic hierarchies and model personalized uncertainty.
• Experimental results show improved accuracy, calibration, and diversity, making it effective in sparse or conceptually abstract domains.

ManifoldMind is a probabilistic geometric recommender system designed for dynamic, trustworthy reasoning over semantic hierarchies in hyperbolic space. The system represents users, items, and semantic tags with adaptive-curvature probabilistic spheres and employs a curvature-aware semantic kernel to enable personalized uncertainty modeling and multi-hop inference. This architecture supports transparent, exploration-driven recommendations, particularly in sparse or conceptually abstract domains, by tracing explicit semantic reasoning paths between users and items.

1. System Architecture

ManifoldMind embeds all entities—users, items, and tags—into a hyperbolic space modeled by the Poincaré ball $\mathbb{D}^n$. Each entity $e$ is parameterized as a probabilistic sphere:

• $\mu_e$: Center of the sphere in $\mathbb{D}^n$
• $r_e > 0$: Radius, encoding epistemic uncertainty
• $\kappa_e < 0$: Sectional curvature, learned per entity and not globally fixed

The recommendation pipeline has two main components: (i) the hyperbolic embedding of entities with adaptive curvature and personalized uncertainty, and (ii) a reasoning layer that constructs and scores semantic paths, allowing traversal of the semantic hierarchy from user to item via concept tags. Each inference step employs a beam search procedure to explore multi-hop semantic paths efficiently.

2. Adaptive-Curvature Probabilistic Spheres

A distinctive feature is the assignment of a separately learned negative curvature ($\kappa_e$) to each entity, allowing for localized geometric adaptation. In ManifoldMind, broader and more general concepts are embedded in regions with lower $|\kappa|$ (flatter), while specific, fine-grained entities occupy regions with sharper curvature. For any entity, the triplet $(\mu_e, r_e, \kappa_e)$ jointly captures the entity's main semantic “location,” its uncertainty, and the geometry of its semantic neighborhood.

When comparing two entities $e_i$ and $e_j$, the system combines curvatures using a harmonic mean weighted by their radii, yielding an effective curvature for pairwise interactions. This supports modeling of semantic depth and hierarchical relationships.

3. Semantic Kernel and Multi-hop Inference

For compositional semantic reasoning, ManifoldMind employs a curvature-aware kernel to compute similarities:

$$K(e_i, e_j) = \exp\left(-\frac{d_{\kappa}(e_i, e_j)^2}{r_i^2 + r_j^2 + \epsilon}\right)$$

where $d_{\kappa}(e_i, e_j)$ is the geodesic distance in hyperbolic space under the effective curvature, and $r_i, r_j$ are the radii representing uncertainty. Higher uncertainty (larger radii) reduces the similarity.

The multi-hop inference procedure computes a recommendation score for item $v$ and user $u$ by searching for the semantic path $\mathcal{P}_{u \rightarrow v}$ with optimal confidence:

$$s(u, v) = \max_{\mathcal{P}} \prod_{(e_i, e_{i+1}) \in \mathcal{P}_{u \rightarrow v}} K(e_i, e_{i+1})$$

Taking logarithms yields the additive form:

$$\log s(u, v) = \max_{\mathcal{P}} \sum_{(e_i, e_{i+1}) \in \mathcal{P}} \log K(e_i, e_{i+1})$$

Beam search (e.g., up to 3 hops) is used to explore semantic paths efficiently. This mechanism allows the model to recommend items not only via the nearest neighbors but by aggregating weak or indirect evidence through intervening semantic tags—a capability especially valuable in sparse or abstract domains.

4. Experimental Results and Performance

ManifoldMind was benchmarked on datasets including Book-Crossing, MIND, GoodBooks-10K, and Avicenna-Syllogism. Key performance outcomes include:

• Ranking Accuracy: NDCG@10 of ≈0.473, a 4.5% improvement over the strongest baselines (e.g., FineRec, LightGCN++).
• Calibration: Expected Calibration Error (ECE) over 11% lower than the best alternative, indicating that predicted confidence aligns well with observed outcomes.
• Diversity: Substantially improved Diversity@10 and lower topic-aware intra-list similarity, facilitated by the multi-hop inference mechanism that increases variety and reduces redundancy in recommendation results.

These improvements are traced to two innovations: adaptive local curvature with personalized uncertainty, and soft compositional reasoning across the semantic hierarchy.

5. Reasoning Traces and Interpretability

A principal benefit of ManifoldMind is its transparent reasoning process. The multi-hop semantic path search explicitly records the sequence of entities (user, tag₁, tag₂, ..., item) and associated kernel similarities leading to a recommendation. Each recommended item can be traced back through the reasoning graph, providing interpretable, justifiable recommendations—for example: “because you enjoyed [Concept A] and [Concept B], both closely linked to this item.” This explicit, human-auditable trace distinguishes ManifoldMind from black-box recommenders.

6. Applications and Implications

ManifoldMind’s methodology is especially effective in:

• Sparse domains: Cold-start scenarios or environments with few direct user-item interactions, where semantic path reasoning leverages available conceptual structure.
• Abstract or high-level settings: News, scientific literature, education, and healthcare, where interpretability and semantic diversity are critical.
• Auditability: Domains requiring transparent, trustworthy recommendations, such as regulatory settings or high-stakes decision-support.

The framework’s integration of uncertainty, geometric reasoning, and explicit semantic traces marks a significant advance in probabilistic geometric recommendation systems. ManifoldMind’s blend of adaptive hyperbolic geometry and symbolic inference illustrates the trend toward combining neural and symbolic methods for more robust, interpretable, and user-aligned machine reasoning.