Related papers: Rectangular knot diagrams classification with deep…
Computer-aided detection has been a research area attracting great interest in the past decade. Machine learning algorithms have been utilized extensively for this application as they provide a valuable second opinion to the doctors.…
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number $c$ grows exponentially with $c$, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings.…
This work develops the global equations of neural networks through stacked piecewise manifolds, fixed-point theory, and boundary-conditioned iteration. Once fixed coordinates and operators are removed, a neural network appears as a…
Deep neural networks have been proven powerful at processing perceptual data, such as images and audio. However for tabular data, tree-based models are more popular. A nice property of tree-based models is their natural interpretability. In…
The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using…
Higher-order graph neural networks (HOGNNs) and the related architectures from Topological Deep Learning are an important class of GNN models that harness polyadic relations between vertices beyond plain edges. They have been used to…
In this paper we present a method using deep learning to compute parametrizations for B-spline curve approximation. Existing methods consider the computation of parametric values and a knot vector as separate problems. We propose to train…
Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a…
Diagrams matter. Unfortunately, the deep learning community has no standard method for diagramming architectures. The current combination of linear algebra notation and ad-hoc diagrams fails to offer the necessary precision to understand…
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite…
In this paper, we study deep diagonal circulant neural networks, that is deep neural networks in which weight matrices are the product of diagonal and circulant ones. Besides making a theoretical analysis of their expressivity, we…
A powerful way to study groups is via their actions on suitable spaces. Classifying spaces for families of subgroups are a type of these spaces, obtained by imposing some strict conditions on the fixed-point sets. We show how in the…
Graph neural networks are becoming increasingly popular in the field of machine learning due to their unique ability to process data structured in graphs. They have also been applied in safety-critical environments where perturbations…
Recently, methods have been developed to accurately predict the testing performance of a Deep Neural Network (DNN) on a particular task, given statistics of its underlying topological structure. However, further leveraging this newly found…
In this paper, we study instances of complex neural networks, i.e. neural netwo rks with complex topologies. We use Self-Organizing Map neural networks whose n eighbourhood relationships are defined by a complex network, to classify handwr…
Deep learning (DL) enables deep neural networks (DNNs) to automatically learn complex tasks or rules from given examples without instructions or guiding principles. As we do not engineer DNNs' functions, it is extremely difficult to…
Classical knots in $\mathbb{R}^3$ can be represented by diagrams in the plane. These diagrams are formed by curves with a finite number of transverse crossings, where each crossing is decorated to indicate which strand of the knot passes…
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to…
We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show…
Untangling ropes, wires, and cables is a challenging task for robots due to the high-dimensional configuration space, visual homogeneity, self-occlusions, and complex dynamics. We consider dense (tight) knots that lack space between…