Sparse differential resultant formulas: between the linear and the nonlinear case

A matrix representation of the sparse differential resultant is the basis for efficient computation algorithms, whose study promises a great contribution to the development and applicability of differential elimination techniques. It is shown how sparse linear differential resultant formulas provide bounds for the order of derivation, even in the nonlinear case, and they also provide (in many cases) the bridge with results in the nonlinear algebraic case

Fast Order Basis and Kernel Basis Computation and Related Problems

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases

Computation of Kullback–Leibler Divergence in Bayesian Networks

Kullback–Leibler divergence KL(p, q) is the standard measure of error when we have a true probability distribution p which is approximate with probability distribution q. Its efficient computation is essential in many tasks, as in approximate computation or as a measure of error when learning a probability. In high dimensional probabilities, as the ones associated with Bayesian networks, a direct computation can be unfeasible. This paper considers the case of efficiently computing the Kullback–Leibler divergence of two probability distributions, each one of them coming from a different Bayesian network, which might have different structures. The paper is based on an auxiliary deletion algorithm to compute the necessary marginal distributions, but using a cache of operations with potentials in order to reuse past computations whenever they are necessary. The algorithms are tested with Bayesian networks from the bnlearn repository. Computer code in Python is provided taking as basis pgmpy, a library for working with probabilistic graphical models.Spanish Ministry of Education and Science under project PID2019-106758GB-C31European Regional Development Fund (FEDER

Introduction to Quantum Information Processing

As a result of the capabilities of quantum information, the science of quantum information processing is now a prospering, interdisciplinary field focused on better understanding the possibilities and limitations of the underlying theory, on developing new applications of quantum information and on physically realizing controllable quantum devices. The purpose of this primer is to provide an elementary introduction to quantum information processing, and then to briefly explain how we hope to exploit the advantages of quantum information. These two sections can be read independently. For reference, we have included a glossary of the main terms of quantum information.Comment: 48 pages, to appear in LA Science. Hyperlinked PDF at http://www.c3.lanl.gov/~knill/qip/prhtml/prpdf.pdf, HTML at http://www.c3.lanl.gov/~knill/qip/prhtm

Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use $\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ operations in $\mathbb{K}$, where $s$ is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and $\omega$ is the exponent of matrix multiplication. The soft-$O$ notation indicates that logarithmic factors in the big-$O$ are omitted while the ceiling function indicates that the cost is $\widetilde{\mathcal{O}}(n^\omega)$ when $s = o(1)$. Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm