Related papers: Invariant Representations of Embedded Simplicial C…

In recent years, there has been a resurgence in methods that use distributed (neural) representations to represent and reason about semantic knowledge for robotics applications. However, while robots often observe previously unknown…

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many…

Graph embedding techniques have attracted growing interest since they convert the graph data into continuous and low-dimensional space. Effective graph analytic provides users a deeper understanding of what is behind the data and thus can…

Recent works in geometric deep learning have introduced neural networks that allow performing inference tasks on three-dimensional geometric data by defining convolution, and sometimes pooling, operations on triangle meshes. These methods,…

3D printing of surfaces has become an established method for prototyping and visualisation. However, surfaces often contain certain degenerations, such as self-intersecting faces or non-manifold parts, which pose problems in obtaining a 3D…

Geometric shape classification of vector polygons remains a challenging task in spatial analysis. Previous studies have primarily focused on deep learning approaches for rasterized vector polygons, while the study of discrete polygon…

In distributed systems, knowledge of the network structure of the connections among the unitary components is often a requirement for an accurate prediction of the emerging collective dynamics. However, in many real-world situations, one…

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links,…

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…

Many disciplines of science and engineering deal with problems related to compositions, ranging from chemical compositions in materials science to portfolio compositions in economics. They exist in non-Euclidean simplex spaces, causing many…

We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.

This paper explores a particular statistical model on 6-valent graphs with special properties which turns out to be invariant with respect to certain Roseman moves if the graph is the singular point graph of a diagram of a 2-knot. The…

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is…

We propose an algorithm to estimate the topology of an embedded metric graph from a well-sampled finite subset of the underlying graph.

Obtaining sparse, interpretable representations of observable data is crucial in many machine learning and signal processing tasks. For data representing flows along the edges of a graph, an intuitively interpretable way to obtain such…

The space of graphs is often characterised by a non-trivial geometry, which complicates learning and inference in practical applications. A common approach is to use embedding techniques to represent graphs as points in a conventional…

Learning low-dimensional numerical representations from symbolic data, e.g., embedding the nodes of a graph into a geometric space, is an important concept in machine learning. While embedding into Euclidean space is common, recent…

Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman \cite{Goodman1949} and Frank \cite{Frank1978}. We revisit a problem formulated by Frank…

This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the…