Knowledge Graph Embedding (KGE)

The foundational paradigm where relations are modeled as vector translations from head to tail entities in the embedding space. TransE defines the score function as ||h + r - t||, enforcing that the head entity vector plus the relation vector approximates the tail entity vector. TransR extends this by projecting entities into a relation-specific space via a learned matrix, capturing diverse relational semantics. These models excel at modeling one-to-one relationships but struggle with complex patterns like symmetry and composition.

Leverage tensor factorization to measure the compositional similarity of entity-relation pairs. RESCAL captures pairwise interactions through a bilinear form h^T M_r t, where each relation has a full matrix, enabling rich modeling at the cost of parameter explosion. DistMult simplifies this to a diagonal matrix, reducing parameters but limiting to symmetric relations only. ComplEx extends DistMult into complex-valued embeddings, where the Hermitian dot product naturally captures antisymmetry, making it highly effective for modeling directed graphs.

Modern KGE methods use message-passing neural networks to aggregate neighborhood information before scoring. R-GCN applies relation-specific weight matrices during convolution, learning entity representations that encode multi-hop relational context. CompGCN jointly embeds entities and relations using composition operators like subtraction, multiplication, and circular correlation. These encoders capture higher-order graph structure beyond direct triples, significantly improving performance on inductive link prediction where entities are unseen during training.

Advanced embedding spaces address fundamental representational limitations of Euclidean geometry. RotatE embeds entities in complex space and defines each relation as a rotation from head to tail, naturally modeling symmetry, antisymmetry, inversion, and composition. MuRP operates in hyperbolic space with negative curvature, ideal for representing hierarchical taxonomies common in retail catalogs. Hyperbolic embeddings achieve lower distortion when encoding tree-like structures, preserving parent-child and sibling relationships with fewer dimensions.

Training KGE models requires specialized loss formulations to distinguish true triples from corrupted negatives. Margin-based ranking loss enforces a minimum separation between positive and negative scores: max(0, γ - f(h,r,t) + f(h',r,t')). Binary cross-entropy with negative sampling treats link prediction as a classification problem, applying a sigmoid to scores. Self-adversarial negative sampling weights negatives by their current model scores, focusing training on the most challenging false triples and accelerating convergence.

The primary downstream application where KGEs predict missing edges in a knowledge graph. In retail, this translates to predicting user-item interactions as a relation like purchased_by or interested_in. The model scores all candidate tail entities for a query (user, purchased_by, ?) and returns the top-K items. This approach naturally handles cold-start items through entity-side features and enables explainable recommendations by surfacing the relational paths connecting a user to a suggested product through intermediate entities like brands, categories, or co-purchased items.