Systematic approaches to the analysis and design of optimization algorithms
Adrien Taylor (INRIA, Paris)
In this talk, I will provide a high-level overview of recent principled approaches for constructively analyzing and designing numerical optimization algorithms. The presentation will be example-based, as the main ingredients necessary for understanding the methodologies are already present in the analysis of base optimization schemes, such as gradient descent. Based on those examples, I will discuss how those techniques can be leveraged for constructing Lyapunov-based analyses and optimal convex optimization algorithms. The methodology can be accessed through easy-to-use open-source packages (including PEPit: https://github.com/PerformanceEstimation/PEPit), allowing the use of the framework without the modelling pain. This talk is based on joint works with great colleagues that I will introduce during the presentation.
Spectral Phases of the Erdős–Rényi graph
Johannes Alt (Bonn University)
We consider the Erdős–Rényi graph on N vertices with expected degree d for each vertex. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of eigenvectors remains delocalized. In this talk, I will explain our results in both phases and present the phase diagram depicting them. For a certain regime in d, we establish a mobility edge by showing that the localized phase extends up to the boundary of the delocalized phase. This is based on joint works with Raphael Ducatez and Antti Knowles.
(at 14:00) Algebraic machine learning and the problem of finding singularities
Markus Pflaum (University of Colorado)
A common assumption, which goes by the name manifold hypothesis asserts that in scientific models data sets with values in $\mathbb{R}^n$ occupy smooth manifolds of dimension less than $n$. The manifold hypothesis fails though in case the underlying geometry of the model contains singularities. Using methods from real algebraic geometry and statistics we present in the first part of this talk a method for learning the underlying variety of a data set in euclidean space. In a further step we explain numerical methods how to find singularities in this variety using Gröbner basis computations. In the second part of the talk we consider energy landscapes from physical chemistry (and other chemically defined surfaces) as potential areas of application. Energy landscapes usually appear as potential energy surfaces of molecules and are usually smooth away from the unphysical points of coinciding atoms. This changes though under reduction of degrees of freedem. The resulting reduced energy landscape possesses in general singularities. The phenomenon will be explained at the example of cyclooctane and possible chemical implications will be indicated. Another situation where singularities in chemistry can appear is as degenerate critical points of important electronic functions such as the electron density distribution or the electron localization function. At such examples it will be explained how modern results from singularity and catastrophe theory could be used to create new topological detectors within chemistry.
TBA
Cordian Riener (UiT The Artic University)
TBA