Related papers: Towards Stratified Space Learning: 2-complexes
Intersection homology is a topological invariant which detects finer information in a space than ordinary homology. Using ideas from classical simple homotopy theory, we construct local combinatorial transformations on simplicial complexes…
Three-dimensional (3D) urban models have gained interest because of their applications in many use-cases such as urban planning and virtual reality. However, generating these 3D representations requires LiDAR data, which are not always…
Models trained on semantically related datasets and tasks exhibit comparable inter-sample relations within their latent spaces. We investigate in this study the aggregation of such latent spaces to create a unified space encompassing the…
We introduce Deep Set Linearized Optimal Transport, an algorithm designed for the efficient simultaneous embedding of point clouds into an $L^2-$space. This embedding preserves specific low-dimensional structures within the Wasserstein…
We study the space $\nua{m}{d}$ of clouds in $\bbr^d$ (ordered sets of $m$ points modulo the action of the group of affine isometries). We show that $\nua{m}{d}$ is a smooth space, stratified over a certain hyperplane arrangement in…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
The problem of learning a manifold structure on a dataset is framed in terms of a generative model, to which we use ideas behind autoencoders (namely adversarial/Wasserstein autoencoders) to fit deep neural networks. From a machine learning…
Recently, neural network architectures have been developed to accommodate when the data has the structure of a graph or, more generally, a hypergraph. While useful, graph structures can be potentially limiting. Hypergraph structures in…
Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that…
Point cloud is an important type of geometric data structure. Due to its irregular format, most researchers transform such data to regular 3D voxel grids or collections of images. This, however, renders data unnecessarily voluminous and…
Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description…
We introduce an information-theoretic framework for smooth structures on topological manifolds, replacing coordinate charts with small-scale entropy data of local probability probes. A concise set of axioms identifies admissible coordinate…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
Abstract separation systems provide a simple general framework in which both tree-shape and high cohesion of many combinatorial structures can be expressed, and their duality proved. Applications range from tangle-type duality and tree…
Point clouds have become increasingly vital across various applications thanks to their ability to realistically depict 3D objects and scenes. Nevertheless, effectively compressing unstructured, high-precision point cloud data remains a…
We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their…
Line clouds, though under-investigated in the previous work, potentially encode more compact structural information of buildings than point clouds extracted from multi-view images. In this work, we propose the first network to process line…
We define a notion of a symplectic structure on stratified spaces, and demonstrate that given a symplectic structure on a stratified space $X$ with integral cohomology class, $X$ can be symplectically embedded in some complex projective…
A stratified space is a topological space equipped with a \emph{stratification}, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e.,…
We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof…