Related papers: Computing the Kreiss Constant of a Matrix
This work is a continuation of "Fast and backward stable computation of roots of polynomials" by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015. In that paper…
This paper presents a fast and powerful method for the computation of eigenvalue bounds for Hessian matrices $\nabla^2 \varphi(x) $ of nonlinear functions $\varphi: U \subseteq R^n\rightarrow R$ on hyperrectangles $B \subset U$. The method…
We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity. We give a novel approach that solves the…
Accounting for the uncertainty in the predictions of modern neural networks is a challenging and important task in many domains. Existing algorithms for uncertainty estimation require modifying the model architecture and training procedure…
Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov's Accelerated Riemannian Gradient algorithm by efficient orthographic retraction…
We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the…
The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input…
Estimating the diagonal entries of a matrix, that is not directly accessible but only available as a linear operator in the form of a computer routine, is a common necessity in many computational applications, especially in image…
We develop a new algorithm for fitting circles that does not have drawbacks commonly found in existing circle fits. Our fit achieves ultimate accuracy (to machine precision), avoids divergence, and is numerically stable even when fitting…
Quantum computation offers a promising alternative to classical computing methods in many areas of numerical science, with algorithms that make use of the unique way in which quantum computers store and manipulate data often achieving…
Gradient-based solvers risk convergence to local optima, leading to incorrect researcher inference. Heuristic-based algorithms are able to ``break free" of these local optima to eventually converge to the true global optimum. However, given…
Matrix clocks are a generalization of the notion of vector clocks that allows the local representation of causal precedence to reach into an asynchronous distributed computation's past with depth $x$, where $x\ge 1$ is an integer.…
We give faster algorithms and improved sample complexities for estimating the top eigenvector of a matrix $\Sigma$ -- i.e. computing a unit vector $x$ such that $x^T \Sigma x \ge (1-\epsilon)\lambda_1(\Sigma)$: Offline Eigenvector…
We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…
Our goal is to efficiently compute low-dimensional latent coordinates for nodes in an input graph -- known as graph embedding -- for subsequent data processing such as clustering. Focusing on finite graphs that are interpreted as uniform…
We introduce in this paper an optimal first-order method that allows an easy and cheap evaluation of the local Lipschitz constant of the objective's gradient. This constant must ideally be chosen at every iteration as small as possible,…
Variational quantum algorithms are promising tools for near-term quantum computers as their shallow circuits are robust to experimental imperfections. Their practical applicability, however, strongly depends on how many times their circuits…
We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq \epsilon\|b\|$ in time: $$\tilde O((n^2+nk^{\omega-1})\log1/\epsilon),$$ where…
Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern…
Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of…