Related papers: Computing the Determinant of a Dense Matrix over Z
We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminants from both a theoretical and an algorithmic perspective. We show that all these parametrizations are birational and prove some results on the…
In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the…
Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly…
Some algorithms for the numerically exact treatment of fermion determinants are summarised. This is not supposed to be a review, rather a concise handbook. The audience is expected to have a basic understanding of how to put fermions on a…
Within ab initio Quantum Monte Carlo simulations, the leading numerical cost for large systems is the computation of the values of the Slater determinants in the trial wavefunction. Each Monte Carlo step requires finding the determinant of…
Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant…
Following several decades of successive algorithmic improvements, works from the 2010s have showed how to compute the Hermite normal form (HNF) of a univariate polynomial matrix within a complexity bound which is essentially that of…
We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics.…
Let $\mathscr{H}$ be a finite-dimensional complex Hilbert space and $\mathscr{D}$ the set of density matrices on $\mathscr{H}$, i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure $u$ on…
There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques. Here, we present a…
We introduce an algorithm which, in the context of nonlinear regression on vector-valued explanatory variables, chooses those combinations of vector components that provide best prediction. The algorithm devotes particular attention to…
The QZ algorithm computes the Schur form of a matrix pencil. It is an iterative algorithm and at some point, it must decide that an eigenvalue has converged and move on with another one. Choosing a criterion that makes this decision is…
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem…
The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of…
We provide new high-accuracy randomized algorithms for solving linear systems and regression problems that are well-conditioned except for $k$ large singular values. For solving such $d \times d$ positive definite system our algorithms…
Logarithms of determinants of large positive definite matrices appear ubiquitously in machine learning applications including Gaussian graphical and Gaussian process models, partition functions of discrete graphical models, minimum-volume…
In this paper, the sparse sensor placement problem for least-squares estimation is considered, and the previous novel approach of the sparse sensor selection algorithm is extended. The maximization of the determinant of the matrix which…
We study the problem of maximizing a monotone submodular function subject to a matroid constraint, and present for it a deterministic non-oblivious local search algorithm that has an approximation guarantee of $1 - 1/e - \varepsilon$ (for…
We present a new algorithm by which the Adomian polynomials can be determined for scalar-valued nonlinear polynomial functional in a Hilbert space. This algorithm calculates the Adomian polynomials without the complicated operations such as…
This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop $O(n)$ algorithms to…