HoloWedge: Quantum Gravity inspired Learning of Hierarchical Embeddings

Hyperbolic embeddings provide a natural representation of partially ordered data by encoding hierarchical relationships geometrically. However, existing learning methods suffer from undesirable features, including concentration of embedded data near the boundary, exclusion of the central region of hyperbolic space, and the necessity of pretraining using an external model. We introduce HoloWedge, a new framework for learning embeddings of hierarchical data inspired by theoretical physics, specifically the holographic approach to quantum gravity that relates the interior and the boundary of hyperbolic space. HoloWedge represents each data point by a subregion of hyperbolic space, a so-called wedge, which is bounded by a minimal-area hypersurface. We construct loss functions from areas of minimal surfaces, which encourage the containment relation between wedges to reflect the partial order within the underlying data. In the holographic model of theoretical physics, areas of minimal surfaces are identified with the entropies of the corresponding wedges. Our experiments demonstrate that our quantum gravity-inspired organizing principle outperforms existing approaches for reconstruction tasks in a moderate number of dimensions, while remaining competitive for link prediction tasks. The resulting embeddings are less concentrated at the boundary and utilize the full hyperbolic space. Moreover, the use of entropic ordering in our loss functions reduces sensitivity to initialization and removes the need for pretraining. Our results show that holography-inspired methods can effectively learn hierarchical representations.