Related papers: Random Sampling in Computational Algebra: Helly Nu…
Clarksons algorithm is a two-staged randomized algorithm for solving linear programs. This algorithm has been simplified and adapted to fit the framework of LP-type problems. In this framework we can tackle a number of non-linear problems…
Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and…
The scenario approach developed by Calafiore and Campi to attack chance-constrained convex programs utilizes random sampling on the uncertainty parameter to substitute the original problem with a representative continuous convex…
We define a violator operator which captures the definition of a minimal Gr\"obner basis of an ideal. This construction places the problem of computing a Gr\"obner basis within the framework of violator spaces, introduced in 2008 by…
We study randomized algorithms for constrained optimization, in abstract frameworks that include, in strictly increasing generality: convex programming; LP-type problems; violator spaces; and a setting we introduce, consistent spaces. Such…
Large scale optimization problems are ubiquitous in machine learning and data analysis and there is a plethora of algorithms for solving such problems. Many of these algorithms employ sub-sampling, as a way to either speed up the…
The "scenario approach" provides an intuitive method to address chance constrained problems arising in control design for uncertain systems. It addresses these problems by replacing the chance constraint with a finite number of sampled…
We derive computationally tractable methods to select a small subset of experiment settings from a large pool of given design points. The primary focus is on linear regression models, while the technique extends to generalized linear models…
Helly's theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If $S$ is a set of $n$ points in $R^d$, we say that $S$ is $(k,G)$-clusterable if it can be partitioned into…
We introduce a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Gr\"unbaum, which itself is a corollary of Helly's theorem.…
Bayesian hierarchical models have been demonstrated to provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models comprise typically a conditionally Gaussian prior model for the unknown, augmented by…
We consider the problem of approximating a given element $f$ from a Hilbert space $\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the…
Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of $n$ functions over a convex…
This paper presents new quadrature rules for functions in a reproducing kernel Hilbert space using nodes drawn by a sampling algorithm known as randomly pivoted Cholesky. The resulting computational procedure compares favorably to previous…
We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly…
We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas…
Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times…
This paper is devoted to the problem of sampling Gaussian fields in high dimension. Solutions exist for two specific structures of inverse covariance : sparse and circulant. The proposed approach is valid in a more general case and…
The Euclidean space notion of convex sets (and functions) generalizes to Riemannian manifolds in a natural sense and is called geodesic convexity. Extensively studied computational problems such as convex optimization and sampling in convex…
Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…