Hyperbolic Distance Matrices

Learning phylogenetic trees as hyperbolic point configurations

We propose a novel method for the inference of phylogenetic trees that utilises point configurations on hyperbolic space as its optimisation landscape. Each taxon corresponds to a point of the point configuration, while the evolutionary distance between taxa is represented by the geodesic distance between their corresponding points. The point configuration is iteratively modified to increase an objective function that additively combines pairwise log-likelihood terms. After convergence, the final tree is derived from the inter-point distances using a standard distance-based method. The objecti…

通过测地线和极限球面投影的双曲切片-Wasserstein距离

Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections

针对双曲空间中概率分布比较方法稀缺且最优传输计算成本高的问题，提出通过沿极限球面或测地线投影构造新的双曲切片-Wasserstein差异度量，并在采样和图像分类任务中验证其有效性。

Strain-Minimizing Hyperbolic Network Embeddings with Landmarks

We introduce L-hydra (landmarked hyperbolic distance recovery and approximation), a method for embedding network- or distance-based data into hyperbolic space, which requires only the distance measurements to a few 'landmark nodes'. This landmark heuristic makes L-hydra applicable to large-scale graphs and improves upon previously introduced methods. As a mathematical justification, we show that a point configuration in d-dimensional hyperbolic space can be perfectly recovered (up to isometry) from distance measurements to just d+1 landmarks. We also show that L-hydra solves a two-stage strain…

Weisfeiler and Leman go Hyperbolic: Learning Distance Preserving Node Representations

In recent years, graph neural networks (GNNs) have emerged as a promising tool for solving machine learning problems on graphs. Most GNNs are members of the family of message passing neural networks (MPNNs). There is a close connection between these models and the Weisfeiler-Leman (WL) test of isomorphism, an algorithm that can successfully test isomorphism for a broad class of graphs. Recently, much research has focused on measuring the expressive power of GNNs. For instance, it has been shown that standard MPNNs are at most as powerful as WL in terms of distinguishing non-isomorphic graphs. …

Hyperbolic Image Embeddings

Computer vision tasks such as image classification, image retrieval and few-shot learning are currently dominated by Euclidean and spherical embeddings, so that the final decisions about class belongings or the degree of similarity are made using linear hyperplanes, Euclidean distances, or spherical geodesic distances (cosine similarity). In this work, we demonstrate that in many practical scenarios hyperbolic embeddings provide a better alternative.

Riemannian Metric Learning for Symmetric Positive Definite Matrices

Over the past few years, symmetric positive definite (SPD) matrices have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the two most widely-used measures are affine-invariant distance and log-Euclidean distance. This is because these two measures are true geodesic distances induced by Riemannian geometry. In this work, we focus on the log-Euclidean Riemannian geometry and propose a data-driven approach for learning Riemannian metrics/geodesic distances for SPD matrices. We show th…