Ron Kimmel

Saar Huberman, Amit Bracha, Ron Kimmel

A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface and estimating distances on the polygon. We show that exact geodesic distances restricted to the polygon are at most second-order accurate with respect to the distances on the corresponding continuous surface. By order of accuracy we refer to the convergence rate as a function of the average distance between sampled points. Next, a higher-order accurate deep learning method for computing geodesic distances on surfaces is introduced. Traditionally, one considers two main components when computing distances on surfaces: a numerical solver that locally approximates the distance function, and an efficient causal ordering scheme by which surface points are updated. Classical minimal path methods often exploit a dynamic programming principle with quasi-linear computational complexity in the number of sampled points. The quality of the distance approximation is determined by the local solver that is revisited in this paper. To improve state of the art accuracy, we consider a neural network-based local solver which implicitly approximates the structure of the continuous surface. We supply numerical evidence that the proposed learned update scheme provides better accuracy compared to the best possible polyhedral approximations and previous learning-based methods. The result is a third-order accurate solver with a bootstrapping-recipe for further improvement.

Thomas Dagès, Simon Weber, Daniel Cremers et al.

Manifold learning is a fundamental task at the core of data analysis and visualisation. It aims to capture the simple underlying structure of complex high-dimensional data by preserving pairwise dissimilarities in low-dimensional embeddings. Traditional methods rely on symmetric Riemannian geometry, thus forcing symmetric dissimilarities and embedding spaces, e.g. Euclidean. However, this discards in practice valuable asymmetric information inherent to the non-uniformity of data samples. We suggest to harness this asymmetry by switching to Finsler geometry, an asymmetric generalisation of Riemannian geometry, and propose a Finsler manifold learning pipeline that constructs asymmetric dissimilarities and embeds in a Finsler space. This greatly broadens the applicability of existing asymmetric embedders beyond traditionally directed data to any data. We also modernise asymmetric embedders by generalising current reference methods to asymmetry, like Finsler t-SNE and Finsler Umap. On controlled synthetic and large real datasets, we show that our asymmetric pipeline reveals valuable information lost in the traditional pipeline, e.g. density hierarchies, and consistently provides superior quality embeddings than their Euclidean counterparts.