Related papers: Unshackling Linear Algebra from Linear Notation
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was…
A thrackle is a drawing of a graph on a surface such that (i) adjacent edges only intersect at their common vertex; and (ii) nonadjacent edges intersect at exactly one point, at which they cross. Conway conjectured that if a graph with $n$…
Finding patterns in graphs is a fundamental problem in databases and data mining. In many applications, graphs are temporal and evolve over time, so we are interested in finding durable patterns, such as triangles and paths, which persist…
Existing network embedding approaches tackle the problem of learning low-dimensional node representations. However, networks can also be seen in the light of edges interlinking pairs of nodes. The broad goal of this paper is to introduce…
With the advent of computers, one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure, namely, n-linear algebras of type I are introduced in this book and its applications to n-Markov chains…
Trellises are crucial graphical representations of codes. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. Iterative decoding concretely motivates such theory. In…
We give some graph theoretical formulas for the trace $Tr_k(\mathbb {T})$ of a tensor $\mathbb {T}$ which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of…
Despite deep learning models running well-defined mathematical functions, we lack a formal mathematical framework for describing model architectures. Ad-hoc notation, diagrams, and pseudocode poorly handle nonlinear broadcasting and the…
A graph is a data structure composed of dots (i.e. vertices) and lines (i.e. edges). The dots and lines of a graph can be organized into intricate arrangements. The ability for a graph to denote objects and their relationships to one…
The methods of the space syntax have been the subject of extensive discussion, and several techniques to identify the axis lines have been proposed. The space syntax can be represented in terms of line graph, a graphs defined on the edge of…
Many problems in machine learning can be expressed by means of a graph with nodes representing training samples and edges representing the relationship between samples in terms of similarity, temporal proximity, or label information. Graphs…
Machine learning on graphs is an important and ubiquitous task with applications ranging from drug design to friendship recommendation in social networks. The primary challenge in this domain is finding a way to represent, or encode, graph…
Graph embedding techniques allow to learn high-quality feature vectors from graph structures and are useful in a variety of tasks, from node classification to clustering. Existing approaches have only focused on learning feature vectors for…
Understanding the notion of a model is not always easy in logic courses. Hence, tools such as Euler diagrams are frequently applied as informal illustrations of set-theoretical models. We formally investigate Euler diagrams as an…
A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for…
The value proposition of a dataset often resides in the implicit interconnections or explicit relationships (patterns) among individual entities, and is often modeled as a graph. Effective visualization of such graphs can lead to key…
This tutorial covers a few recent papers in the field of network embedding. Network embedding is a collective term for techniques for mapping graph nodes to vectors of real numbers in a multidimensional space. To be useful, a good embedding…
Transformers are effective and efficient at modeling complex relationships and learning patterns from structured data in many applications. The main aim of this paper is to propose and design NLAFormer, which is a transformer-based…
A graph is a powerful concept for representation of relations between pairs of entities. Data with underlying graph structure can be found across many disciplines and there is a natural desire for understanding such data better. Deep…
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix…