Related papers: On visualisation scaling, subeigenvectors and Klee…
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…
An effort has been made to show mathematicians some new ideas applied to image analysis. Gray images are presented as tilings. Based on topological properties of the tiling, a number of gray convex hulls: maximal, minimal, and oriented ones…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
This survey highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compresses it to a much smaller matrix by multiplying it by a…
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…
An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix cube problems whose constraints are the minimum and maximum eigenvalue function on an affine space of symmetric…
In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property…
Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer…
We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of $f(a^\top x)$,…
In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…
In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $\lambda B-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to "balance" in some sense the…
We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that these equations can be solved using the Frobenius trace down method in a way similar to…
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
Supervised classification and representation learning are two widely used classes of methods to analyze multivariate images. Although complementary, these methods have been scarcely considered jointly in a hierarchical modeling. In this…
Inspired by recent work on convex formulations of clustering (Lashkari & Golland, 2008; Nowozin & Bakir, 2008) we investigate a new formulation of the Sparse Coding Problem (Olshausen & Field, 1997). In sparse coding we attempt to…
We introduce the concepts of max-closedness and numeraires of convex subsets in the nonnegative orthant of the topological vector space of all random variables built over a probability space, equipped with a topology consistent with…
We study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense…