Related papers: Fast and Near-Optimal Diagonal Preconditioning
Preconditioning has long been a staple technique in optimization, often applied to reduce the condition number of a matrix and speed up the convergence of algorithms. Although there are many popular preconditioning techniques in practice,…
We study optimal diagonal preconditioning using the classical worst-case $\kappa$-condition number and the averaging-based $\omega$-condition number. For the $\kappa$-optimal preconditioning problem, we derive an affine-based pseudoconvex…
We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including…
We consider the problem of finding the optimal diagonal preconditioner for a positive definite matrix. Although this problem has been shown to be solvable and various methods have been proposed, none of the existing approaches are scalable…
We present a new class of preconditioned iterative methods for solving linear systems of the form $Ax = b$. Our methods are based on constructing a low-rank Nystr\"om approximation to $A$ using sparse random matrix sketching. This…
Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. So, while solving a sequence of many linear systems,…
To precondition a large and sparse linear system, two direct methods for approximate factoring of the inverse are devised. The algorithms are fully parallelizable and appear to be more robust than the iterative methods suggested for the…
Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g.,in quasi-Newton methods. Motivated by the latter, we study…
There has been a growing interest in parallel strategies for solving trajectory optimization problems. One key step in many algorithmic approaches to trajectory optimization is the solution of moderately-large and sparse linear systems.…
The main computational cost of algorithms for computing reduced-order models of parametric dynamical systems is in solving sequences of very large and sparse linear systems. We focus on efficiently solving these linear systems, arising…
We address the slow convergence and poor stability of quasi-newton sequential quadratic programming (SQP) methods that is observed when solving experimental design problems, in particular when they are large. Our findings suggest that this…
Recent literature has advocated the use of randomized methods for accelerating the solution of various matrix problems arising throughout data science and computational science. One popular strategy for leveraging randomization is to use it…
We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…
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…
The hierarchical interpolative factorization for elliptic partial differential equations is a fast algorithm for approximate sparse matrix inversion in linear or quasilinear time. Its accuracy can degrade, however, when applied to strongly…
Interior point methods solve small to medium sized problems to high accuracy in a reasonable amount of time. However, for larger problems as well as stochastic problems, one needs to use first-order methods such as stochastic gradient…
We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but…
Modern adaptive optimization methods, such as Adam and its variants, have emerged as the most widely used tools in deep learning over recent years. These algorithms offer automatic mechanisms for dynamically adjusting the update step based…
Although some preconditioners are available for solving dense linear systems, there are still many matrices for which preconditioners are lacking, in particular in cases where the size of the matrix $N$ becomes very large. There remains…
We study linear preconditioning in Markov chain Monte Carlo. We consider the class of well-conditioned distributions, for which several mixing time bounds depend on the condition number $\kappa$. First we show that well-conditioned…