Conference Agenda

Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefinite programming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory.

The theme of the third session is "Numerical methods in line configurations and spectral decompositions."

The problem of equiangular lines asks for the maximum number of lines in the n-dimensional Euclidean space with a fixed common angle. By selecting a unit vector along each line we get a spherical code with inner-products a and -a for some fixed 0 < a < 1 and by making a graph with these vectors as vertices and defining an edge whenever the inner-product is -a, this problem also becomes interesting from the algebraic graph theory point of view.

Parallel to this, techniques from semidefinite programming have been successfully applied to many problems in discrete geometry. The problem is modeled with an infinite graph and methods such as the Lovász theta-number used to upper bound the independence number of a finite graph are extended to this infinite setting. A key step in this approach is the use of symmetries from the problem to simplify the formulation and make it solvable by computer.

In this work we define a hierarchy of semidefinite programs specially suited for symmetry reduction techniques. Let S^{n-1} denote the unit sphere in R^n. In the setting of spherical codes, the symmetry group is the orthogonal group O(n) and we consider kernels S^{n-1} x S^{n-1} -> R invariant with respect to the stabilizer subgroup of a set of k points. We show that these kernels can be represented with matrices with size depending on k and not on n and use this characterization to compute new bounds for the maximum number of equiangular lines with fixed common angle.

Joint work with D. de Laat (MIT), F.M. de Oliveira Filho (TU Delft), and F. Vallentin (Universität zu Köln).

Quantum information has seen a very rapid development in the recent years mostly because it provides very abstract and, at the same time, intuitive view of the quantum theory. It makes it easier for us to find new applications in a wide range of fields from metrology, through cryptography to pure mathematics. In this talk I will start by briefly explaining why and how this happens and then spend the most of the time by presenting an example: our recent result in which we were able to link the number of mutually unbiased bases (MUBs) in any given dimension with a success probability of a certain information-processing task. Next I will present how to bound this probability with a hierarchy of semi-definite programmes (SDPs) and report on how we applied it in practice and where did it lead us.

Many geometrical and combinatorial problems can be formulated as the minimization of specific kernel functions over compact subsets. To derive efficient numerical schemes, we study the spectral decomposition of discrepancy kernels. In particular, we obtain the kernels' Fourier expansions for several distinct examples.

This talk considers the detection of ambiguities that arise in the reconstruction of signals that are measured by a linear array which is given by a set of sensors located on a line in the plane.

In this case, signals are related to measurements by a linear equation system, whose matrix depends on the directions-of-arrival (DOA) of the signals and the positions of the sensors.

It can happen that certain DOAs together with certain sensor positions yield a measurement matrix that does not have full rank. This means that for a corresponding measurement the underlying original signal cannot be uniquely identified, and an ambiguity is said to arise.

In order to detect which DOAs produce ambiguities, we consider quadratic submatrices of the measurement matrix. Using Young tableaux and roots of unity, determining rank-deficient quadratic submatrices can be reduced to a mixed-integer program, whose solutions correspond to roots of the so-called Schur polynomial and thus to ambiguities. We demonstrate this approach using examples.