  In large-scale recommender systems, the user-item networks are generally
scale-free or expand exponentially. The latent features (also known as
embeddings) used to describe the user and item are determined by how well the
embedding space fits the data distribution. Hyperbolic space offers a spacious
room to learn embeddings with its negative curvature and metric properties,
which can well fit data with tree-like structures. Recently, several hyperbolic
approaches have been proposed to learn high-quality representations for the
users and items. However, most of them concentrate on developing the hyperbolic
similitude by designing appropriate projection operations, whereas many
advantageous and exciting geometric properties of hyperbolic space have not
been explicitly explored. For example, one of the most notable properties of
hyperbolic space is that its capacity space increases exponentially with the
radius, which indicates the area far away from the hyperbolic origin is much
more embeddable. Regarding the geometric properties of hyperbolic space, we
bring up a Hyperbolic Regularization powered Collaborative Filtering(HRCF) and
design a geometric-aware hyperbolic regularizer. Specifically, the proposal
boosts optimization procedure via the root alignment and origin-aware penalty,
which is simple yet impressively effective. Through theoretical analysis, we
further show that our proposal is able to tackle the over-smoothing problem
caused by hyperbolic aggregation and also brings the models a better
discriminative ability. We conduct extensive empirical analysis, comparing our
proposal against a large set of baselines on several public benchmarks. The
empirical results show that our approach achieves highly competitive
performance and surpasses both the leading Euclidean and hyperbolic baselines
by considerable margins.
