Related papers: The Algebra of Two Dimensional Patterns
In several domains, data objects can be decomposed into sets of simpler objects. It is then natural to represent each object as the set of its components or parts. Many conventional machine learning algorithms are unable to process this…
We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.
Encoding the electronic structure of molecules using 2-electron reduced density matrices (2RDMs) as opposed to many-body wave functions has been a decades-long quest as the 2RDM contains sufficient information to compute the exact molecular…
Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide…
Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from…
We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…
We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solutions can be represented…
Given a sequence of finite element spaces which form a de Rham sequence, we will construct a dual representation of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence.…
This paper consists of a description of the variety of two dimensional associative algebras within the framework of Nonstandard Analysis. By decomposing each algebra in A^2 as sum of a Jordan algebra and a Lie algebra, we calculate the…
We investigate a class of combinatory algebras, called ribbon combinatory algebras, in which we can interpret both the braided untyped linear lambda calculus and framed oriented tangles. Any reflexive object in a ribbon category gives rise…
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…
Recent work in machine learning shows that deep neural networks can be used to solve a wide variety of inverse problems arising in computational imaging. We explore the central prevailing themes of this emerging area and present a taxonomy…
Deep learning has received much attention lately due to the impressive empirical performance achieved by training algorithms. Consequently, a need for a better theoretical understanding of these problems has become more evident in recent…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
We define a special matrix multiplication among a special subset of $2N\x 2N$ matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative…
We introduce Residue Hyperdimensional Computing, a computing framework that unifies residue number systems with an algebra defined over random, high-dimensional vectors. We show how residue numbers can be represented as high-dimensional…
A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel…
This paper introduces and investigates some properties of algebras constructed from the algebra of polynomials via derivation and integration operators using a process presented by Dzhumadildaev in a previous work. In particular, we…
Transformers can generate predictions in two approaches: 1. auto-regressively by conditioning each sequence element on the previous ones, or 2. directly produce an output sequences in parallel. While research has mostly explored upon this…
In this paper we give classification of two-dimensional real evolution algebras. For several chains of evolution algebras we study their classification dynamics.