Related papers: Learning 3-Manifold Triangulations
To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an…
We provide an introduction to the signature method, focusing on its theoretical properties and machine learning applications. Our presentation is divided into two parts. In the first part, we present the definition and fundamental…
Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a…
In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…
Recently, the application of machine learning models has gained momentum in natural sciences and engineering, which is a natural fit due to the abundance of data in these fields. However, the modeling of physical processes from simulation…
Manifold learning flows are a class of generative modelling techniques that assume a low-dimensional manifold description of the data. The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable…
Polygonal meshes provide an efficient representation for 3D shapes. They explicitly capture both shape surface and topology, and leverage non-uniformity to represent large flat regions as well as sharp, intricate features. This…
We demonstrate the triangulability of compact 3-dimensional topological pseudomanifolds and study the properties of such triangulations, including the Hauptvermutung and relations by Alexander star moves and Pachner bistellar moves. We also…
Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth of a compact, connected 3-manifold $M$ defined by \[ \operatorname{tw}(M) =…
We use persistent homology along with the eigenfunctions of the Laplacian to study similarity amongst triangulated 2-manifolds. Our method relies on studying the lower-star filtration induced by the eigenfunctions of the Laplacian. This…
Persistence diagrams (PDs), often characterized as sets of death and birth of homology class, have been known for providing a topological representation of a graph structure, which is often useful in machine learning tasks. Prior works rely…
We study trisections of 4-manifolds obtained by spinning and twist-spinning 3-manifolds, and we show that, given a (suitable) Heegaard diagram for the 3-manifold, one can perform simple local modifications to obtain a trisection diagram for…
Manifold learning approaches seek the intrinsic, low-dimensional data structure within a high-dimensional space. Mainstream manifold learning algorithms, such as Isomap, UMAP, $t$-SNE, Diffusion Map, and Laplacian Eigenmaps do not use data…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…
The information geometry of the 2-manifold of gamma probability density functions provides a framework in which pseudorandom number generators may be evaluated using a neighbourhood of the curve of exponential density functions. The process…
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary…
Graph neural network (GNN) is a popular tool to learn the lower-dimensional representation of a graph. It facilitates the applicability of machine learning tasks on graphs by incorporating domain-specific features. There are various options…
Traditional 3D face models learn a latent representation of faces using linear subspaces from limited scans of a single database. The main roadblock of building a large-scale face model from diverse 3D databases lies in the lack of dense…
Magnetic moments near zigzag edges in graphene allow complex nanostructures with customised spin properties to be realised. However, computational costs restrict theoretical investigations to small or perfectly periodic structures. Here we…
This paper proposes a new neural network based on SPD manifold learning for skeleton-based hand gesture recognition. Given the stream of hand's joint positions, our approach combines two aggregation processes on respectively spatial and…