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The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…
We represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and…
Learning representations of data is an important problem in statistics and machine learning. While the origin of learning representations can be traced back to factor analysis and multidimensional scaling in statistics, it has become a…
Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial…
Given a quadratic two-parameter matrix polynomial Q, we develop a systematic approach to generating a vector space of linear two-parameter matrix polynomials. We identify a set of linearizations of Q that lie in the vector space. Finally,…
Reasoning on knowledge graphs is a challenging task because it utilizes observed information to predict the missing one. Particularly, answering complex queries based on first-order logic is one of the crucial tasks to verify learning to…
We explore a matrix-space model, that is a natural extension to the vector space model for Information Retrieval. Each document can be represented by a matrix that is based on document extracts (e.g. sentences, paragraphs, sections). We…
We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible…
Probabilistic theory and differential equation are powerful tools for the interpretability and guidance of the design of machine learning models, especially for illuminating the mathematical motivation of learning latent variable from…
We assess the situation of our elementary Linear Algebra classes in the US holistically and through personal history recollections. Possible remedies for our elementary Linear Algebra's teaching problems are discussed and a change from…
In this work, we conduct a joint analysis of both Vector Space and Language Models for IR using the mathematical framework of Quantum Theory. We shed light on how both models allocate the space of density matrices. A density matrix is shown…
This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field.…
Presented here is a matrix inversion method utilizing quantum searching algorithm. In this method, huge Hilbert space as a whole spanned by myriad of eigen states is searched and evaluated efficiently by sequential reduction in dimension…
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…
The transition matrix, frequently abbreviated as T-matrix, contains the complete information in a linear approximation of how a spatially localized object scatters an incident field. The T-matrix is used to study the scattering response of…
We study scalar-linear and vector-linear solutions to the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters. These bounds improve…
In this paper, we propose a new family of graphs, matrix graphs, whose vertex set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two…
Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they…