A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings

Personalized item recommendation typically suffers from data sparsity, which is most often addressed by learning vector representations of users and items via low-rank matrix factorization. While this effectively densifies the matrix by assuming users and movies can be represented by linearly dependent latent features, it does not capture more complicated interactions. For example, vector representations struggle with set-theoretic relationships, such as negation and intersection, e.g. recommending a movie that is “comedy and action, but not romance”. In this work, we formulate the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent. To capture this set-theoretic dependence we represent each user and attribute by a hyperrectangle or box (i.e. a Cartesian product of intervals). Box embeddings can intuitively be understood as trainable Venn diagrams, and thus not only inherently represent similarity (via the Jaccard index), but also naturally and faithfully support arbitrary set-theoretic relationships. Queries involving set-theoretic constraints can be efficiently computed directly on the embedding space by performing geometric operations on the representations. We empirically demonstrate the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30% overall.

Online platforms like Netflix and Amazon recommend items—such as movies, products, or travel destinations—by learning from user preferences. Often, users also want to filter results using specific conditions, such as “funny action movies without clowns.” Traditional recommendation systems struggle with these kinds of structured queries, especially when combining multiple preferences or exclusions.Our research introduces a new geometric approach to recommendation that supports such complex user queries. We represent users and item attributes (like “comedy” or “children’s”) as multi-dimensional boxes. These box-shaped regions make it easier to perform logical operations—like intersections and differences—on preferences. This means the system can handle queries that combine conditions (e.g., action ∧ comedy ∧ ¬romance) more naturally than existing methods.We also design efficient ways to compute these combinations and show that our method outperforms standard vector-based approaches in accuracy. While we train our system using only simple interactions (such as user–item or attribute–item pairs), it generalizes well to more complex combinations at test time. Our work bridges the gap between personalized recommendations and structured user intent, enabling more flexible and interpretable recommendation systems.