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.

Phylogenetic inference, grounded in molecular evolution models, is essential for understanding the evolutionary relationships in biological data. Accounting for the uncertainty of phylogenetic tree variables, which include tree topologies and evolutionary distances on branches, is crucial for accurately inferring species relationships from molecular data and tasks requiring variable marginalization. Variational Bayesian methods are key to developing scalable, practical models; however, it remains challenging to conduct phylogenetic inference without restricting the combinatorially vast number of possible tree topologies. In this work, we introduce a novel, fully differentiable formulation of phylogenetic inference that leverages a unique representation of topological distributions in continuous geometric spaces. Through practical considerations on design spaces and control variates for gradient estimations, our approach, GeoPhy, enables variational inference without limiting the topological candidates. In experiments using real benchmark datasets, GeoPhy significantly outperformed other approximate Bayesian methods that considered whole topologies.

Label-free alignment between datasets collected at different times, locations, or by different instruments is a fundamental scientific task. Hyperbolic spaces have recently provided a fruitful foundation for the development of informative representations of hierarchical data. Here, we take a purely geometric approach for label-free alignment of hierarchical datasets and introduce hyperbolic Procrustes analysis (HPA). HPA consists of new implementations of the three prototypical Procrustes analysis components: translation, scaling, and rotation, based on the Riemannian geometry of the Lorentz model of hyperbolic space. We analyze the proposed components, highlighting their useful properties for alignment. The efficacy of HPA, its theoretical properties, stability and computational efficiency are demonstrated in simulations.

While numerous Video Violence Detection (VVD) methods have focused on representation learning in Euclidean space, they struggle to learn sufficiently discriminative features, leading to weaknesses in recognizing normal events that are visually similar to violent events (i.e., ambiguous violence). In contrast, hyperbolic representation learning, renowned for its ability to model hierarchical and complex relationships between events, has the potential to amplify the discrimination between visually similar events. Inspired by these, we develop a novel Dual-Space Representation Learning (DSRL) method for weakly supervised VVD to utilize the strength of both Euclidean and hyperbolic geometries, capturing the visual features of events while also exploring the intrinsic relations between events, thereby enhancing the discriminative capacity of the features. DSRL employs a novel information aggregation strategy to progressively learn event context in hyperbolic spaces, which selects aggregation nodes through layer-sensitive hyperbolic association degrees constrained by hyperbolic Dirichlet energy. Furthermore, DSRL attempts to break the cyber-balkanization of different spaces, utilizing cross-space attention to facilitate information interactions between Euclidean and hyperbolic space to capture better discriminative features for final violence detection. Comprehensive experiments demonstrate the effectiveness of our proposed DSRL.

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, firstly because of the absence of corresponding hyperbolic neural network layers.

This allows us to learn high-precision embeddings without using BigFloats, and enables us to store the resulting embeddings with fewer bits.