The Algebraic Boundary of Graph Elliptopes

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Abstract
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Correlation matrices arise naturally in statistics, machine learning, and many optimization problems. Often, only some pairwise correlations between random variables are known, raising the question of whether the missing correlations can be chosen so that all observed ones are consistent with a single correlation matrix. The set of all feasible partially specified correlation matrices is described by convex geometric objects called graph elliptopes. These objects are also well studied because they arise naturally as semidefinite programming relaxations of the maximum cut problem, yielding the best known approximation algorithm for this NP-hard problem. We investigate the geometric structure of graph elliptopes by studying the polynomial equations describing their boundary. In particular, we completely characterize these equations when the graph is cycle completable. As an application, we give a geometric characterization of when a graph elliptope is itself the feasible region of a semidefinite program, showing that this happens exactly when the graph is chordal. This is based on joint work with Monique Laurent and Simon Telen.

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About the Speaker
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I am a Ph.D. student in Mathematics at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, where I work in the Numerical Algebraic Geometry group under the supervision of Simon Telen. I am part of the TENORS network as Doctoral Candidate 6 (DC6) and I am currently completing a secondment at CWI in Amsterdam under the supervision of Monique Laurent. My research interests lie in applied algebraic geometry (for example, dynamical systems) and algebraic optimization. I am currently focusing on applications in semidefinite programming.