Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Space
This addresses the challenge of efficient classification for hierarchically organized data in hyperbolic spaces, offering a more stable optimization approach, though it appears incremental as it builds on existing large margin classifiers.
The paper tackles the problem of non-convex optimization in large margin classification for hyperbolic spaces by proposing a classifier with horospherical decision boundaries, which results in a geodesically convex optimization problem that guarantees a globally optimal solution and shows competitive performance compared to state-of-the-art methods.
Hyperbolic spaces have been quite popular in the recent past for representing hierarchically organized data. Further, several classification algorithms for data in these spaces have been proposed in the literature. These algorithms mainly use either hyperplanes or geodesics for decision boundaries in a large margin classifiers setting leading to a non-convex optimization problem. In this paper, we propose a novel large margin classifier based on horospherical decision boundaries that leads to a geodesically convex optimization problem that can be optimized using any Riemannian gradient descent technique guaranteeing a globally optimal solution. We present several experiments depicting the competitive performance of our classifier in comparison to SOTA.