Related papers: Solvable Matrix Models
A survey of the interrelationships between matrix models and field theories on the noncommutative torus is presented. The discretization of noncommutative gauge theory by twisted reduced models is described along with a rigorous definition…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
Interesting physical results can be obtained from sigma models by taking the number of fields N to zero. I discuss how one can make sense of this limit by using exact S matrix techniques. I review how this can be done for the case of…
This work explores fundamental modeling and algorithmic issues arising in the well-established MapReduce framework. First, we formally specify a computational model for MapReduce which captures the functional flavor of the paradigm by…
We present a novel method for approximately equilibrating a matrix $A \in {\bf R}^{m \times n}$ using only multiplication by $A$ and $A^T$. Our method is based on convex optimization and projected stochastic gradient descent, using an…
We introduce and compare new compression approaches to obtain regularized solutions of large linear systems which are commonly encountered in large scale inverse problems. We first describe how to approximate matrix vector operations with a…
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable…
The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the…
We consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order…
The graded reflection equation is investigated for the $U_{q}[osp(r|2m)^{(1)}]$ vertex model. We have found four classes of diagonal solutions with at the most one free parameter and twelve classes of non-diagonal ones with the number of…
We consider the problem of matrix completion on an $n \times m$ matrix. We introduce the problem of Interpretable Matrix Completion that aims to provide meaningful insights for the low-rank matrix using side information. We show that the…
In this paper a novel modification of the multilevel Monte Carlo approach, allowing for further significant complexity reduction, is proposed. The idea of the modification is to use the method of control variates to reduce variance at level…
Dimensionality Reduction plays a pivotal role in improving feature learning accuracy and reducing training time by eliminating redundant features, noise, and irrelevant data. Nonnegative Matrix Factorization (NMF) has emerged as a popular…
In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schr\"{o}dinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by…
The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable to…
We consider optimization problems involving the multiplication of variable matrices to be selected from a given family, which might be a discrete set, a continuous set or a combination of both. Such nonlinear, and possibly discrete,…
We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on…
This paper discusses the large N limit of the two-Hermitian-matrix model in zero dimensions, using the hidden BRST method. A system of integral equations previously found is solved, showing that it contained the exact solution of the model…
In this paper we provide an analytical procedure which leads to a system of $(n-2)^2$ polynomial equations whose solutions give the parameterisation of the complex $n\times n$ Hadamard matrices. It is shown that in general the Hadamard…
The multigrid methodology is reviewed. By integrating numerical processes at all scales of a problem, it seeks to perform various computational tasks at a cost that rises as slowly as possible as a function of $n$, the number of degrees of…