Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

Bhuiyan Sanjid Shafique, Ashmal Vayani, Muhammad Maaz, Hanoona Abdul Rasheed, Dinura Dissanayake, Mohammed Irfan Kurpath, Yahya Hmaiti, Go Inoue, Jean Lahoud, Md. Safirur Rashid, Shadid Intisar Quasem, Maheen Fatima, Franco Vidal, Mykola Maslych, Ketan Pravin More, Sanoojan Baliah, Hasindri Watawana, Yuhao Li, Fabian Farestam, Leon Schaller, Roman Tymtsiv, Simon Weber, Hisham Cholakkal, Ivan Laptev, Shin’ichi Satoh, Michael Felsberg, Mubarak Shah, Salman Khan, Fahad Shahbaz Khan. Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing. 2025.

The Laplace-Beltrami operator (LBO) emerges from studying manifolds equipped with a Riemannian metric. It is often called the swiss army knife of geometry processing as it allows to capture intrinsic shape information and gives rise to heat diffusion, geodesic distances, and a mul-titude of shape descriptors. It also plays a central role in geometric deep learning. In this work, we explore Finsler manifolds as a generalization of Riemannian manifolds. We revisit the Finsler heat equation and derive a Finsler heat kernel and a Finsler-Laplace-Beltrami Operator (FLBO): a novel theoretically justified anisotropic Laplace-Beltrami operator (ALBO). In experimental evaluations we demon-strate that the proposed FLBO is a valuable alternative to the traditional Riemannian-based LBO and ALBOs for spa-tialfiltering and shape correspondence estimation. We hope that the proposed Finsler heat kernel and the FLBO will inspire further exploration of Finsler geometry in the Computer vision community.

Hierarchy is a natural representation of semantic taxonomies, including the ones routinely used in image segmentation. Indeed, recent work on semantic segmentation reports improved accuracy from supervised training leveraging hierarchical label structures. Encouraged by these results, we revisit the fundamental assumptions behind that work. We postulate and then empirically verify that the reasons for the observed improvement in segmentation accuracy may be entirely unrelated to the use of the semantic hierarchy. To demonstrate this, we design a range of crossdomain experiments with a representative hierarchical approach. We find that on the new testing domains, a flat (non-hierarchical) segmentation network, in which the parents are inferred from the children, has superior segmentation accuracy to the hierarchical approach across the board. Complementing these findings and inspired by the intrinsic properties of hyperbolic spaces, we study a more principled approach to hierarchical segmentation using the Poincare ball model. The hyperbolic representation largely outperforms the previous (Euclidean) hierarchical approach as well and is on par with our flat Euclidean baseline in terms of segmentation accuracy. However, it additionally exhibits surprisingly strong calibration quality of the parent nodes in the semantic hierarchy, especially on the more challenging domains. Our combined analysis suggests that the established practice of hierarchical segmentation may be limited to in-domain settings, whereas flat classifiers generalize substantially better, especially if they are modeled in the hyperbolic space.

We introduce Power Bundle Adjustment as an expansion type algorithm for solving large-scale bundle adjustment problems. It is based on the power series expansion of the inverse Schur complement and constitutes a new family of solvers that we call inverse expansion methods. We theoretically justify the use of power series and we prove the convergence of our approach. Using the real-world BAL dataset we show that the proposed solver challenges the state-of-the-art iterative methods and significantly accelerates the solution of the normal equation, even for reaching a very high accuracy. This easy-to-implement solver can also complement a recently presented distributed bundle adjustment framework. We demonstrate that employing the proposed Power Bundle Adjustment as a subproblem solver significantly improves speed and accuracy of the distributed optimization.