Algorithms for high dimensional stabbing problems

Approximation Algorithms for Rectangle Stabbing and Interval Stabbing Problems

In the weighted rectangle stabbing problem we are given a grid in ]R2 consisting of columns and rows each having a positive integral weight, and a set of closed axis-parallel rectangles each having a positive integral demand. The rectangles are placed arbitrarily in the grid with the only assumption that each rectangle is intersected by at least one column and at least one row. The objective is...

Better Algorithms for High-dimensional Proximity Problems via Asymmetri Embeddings

Greedy Algorithms for Structurally Constrained High Dimensional Problems

A hallmark of modern machine learning is its ability to deal with high dimensional problems by exploiting structural assumptions that limit the degrees of freedom in the underlying model. A deep understanding of the capabilities and limits of high dimensional learning methods under specific assumptions such as sparsity, group sparsity, and low rank has been attsined. Efforts [1,2] are now under...

Sparse Approximation Algorithms for High Dimensional Parametric Initial Value Problems

We consider the efficient numerical approximation on nonlinear systems of initial value Ordinary Differential Equations (ODEs) on Banach state spaces S over R or C. We assume the right hand side depends in affine fashion on a vector y = (yj)j≥1 of possibly countably many parameters, normalized such that |yj | ≤ 1. Such affine parameter dependence of the ODE arises, among others, in mass action ...

Greedy Algorithms for High-dimensional Non-symmetric Linear Problems

In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor product functions, each term of which is iteratively computed via a greedy algorithm [20]. There exists a good theoretical framework for these methods in th...