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

TL;DR

This paper introduces a geometric set-theoretic approach to personalized recommendation using box embeddings, effectively capturing complex relationships and outperforming traditional vector-based methods by up to 30%.

Contribution

It proposes representing users and attributes as box embeddings to naturally encode set-theoretic relationships in recommendation systems.

Findings

Box embeddings outperform vector-based methods by up to 30%.

Set-theoretic constraints can be efficiently handled through geometric operations.

The approach captures complex user-item interactions beyond linear dependencies.

Abstract

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 hyper-rectangle 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.