[BOOK][B] Stochastic and integral geometry
R Schneider, W Weil - 2008 - Springer
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Sparse nonnegative solution of underdetermined linear equations by linear programming
Consider an underdetermined system of linear equations y= Ax with known y and d× n
matrix A. We seek the nonnegative x with the fewest nonzeros satisfying y= Ax. In general …
matrix A. We seek the nonnegative x with the fewest nonzeros satisfying y= Ax. In general …
Neighborliness of randomly projected simplices in high dimensions
Let A be ad× n matrix and T= Tn-1 be the standard simplex in R n. Suppose that d and n are
both large and comparable: d≈ δ n, δ∈(0, 1). We count the faces of the projected simplex …
both large and comparable: d≈ δ n, δ∈(0, 1). We count the faces of the projected simplex …
Basic properties of convex polytopes
Convex polytopes are fundamental geometric objects that have been investigated since
antiquity. The beauty of their theory is nowadays complemented by their importance for …
antiquity. The beauty of their theory is nowadays complemented by their importance for …
Counting the faces of randomly-projected hypercubes and orthants, with applications
Let A be an n× N real-valued matrix with n< N; we count the number of k-faces fk (AQ) when
Q is either the standard N-dimensional hypercube IN or else the positive orthant ℝ+ N. To …
Q is either the standard N-dimensional hypercube IN or else the positive orthant ℝ+ N. To …
Central limit theorems for Gaussian polytopes
Choose n random, independent points in R d according to the standard normal distribution.
Their convex hull K n is the Gaussian random polytope. We prove that the volume and the …
Their convex hull K n is the Gaussian random polytope. We prove that the volume and the …
Approximation of convex sets by polytopes
EM Bronstein - Journal of Mathematical Sciences, 2008 - Springer
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Random polytopes
D Hug - Stochastic Geometry, Spatial Statistics and Random …, 2012 - Springer
Random polytopes arise naturally as convex hulls of random points selected according to a
given distribution. In a dual way, they can be derived as intersections of random halfspaces …
given distribution. In a dual way, they can be derived as intersections of random halfspaces …
Thresholds for the recovery of sparse solutions via l1 minimization
The ubiquitous least squares method for systems of linear equations returns solutions which
typically have all non-zero entries. However, solutions with the least number of non-zeros …
typically have all non-zero entries. However, solutions with the least number of non-zeros …
Angles of random simplices and face numbers of random polytopes
Z Kabluchko - Advances in Mathematics, 2021 - Elsevier
Pick d+ 1 points uniformly at random on the unit sphere in R d. What is the expected value of
the angle sum of the simplex spanned by these points? Choose n points uniformly at …
the angle sum of the simplex spanned by these points? Choose n points uniformly at …