Quantum Topology Seminar
Nima Dehmamy
Center for Science of Science and Innovation (CSSI), the Northwestern Institute on Complex Systems
Using knot theory and topology to understand rough energy landscapes
Abstract: Dynamics and optimization problems on many complex systems (i.e. with high number of random interactions) exhibit a multitude of critical points. Examples include energy minima of spin-glasses, protein folding and reaction-diffusion on random graphs. The rough energy landscape of such systems has posed a major challenge in making theoretical advancements about finding optimal solutions. Yet, despite the existence of an intractably large number of local minima, many heuristic algorithms such as gradient descent often find reasonably good local minima. The big question that arises is why and when they can be expected to perform well. I will present two projects in which we tackle aspects of this problem. In both we use the problem of embedding graphs (a type of reaction-diffusion) as a model to study rough energy landscapes. For 3D embedding of graphs with elastic edges, we show that tools from knot theory, such as the linking number, can reveal a lot about the structure of the energy landscape. In particular, we find that the linking number of a graph embedding is approximately proportional to the elastic energy of the edges. We then prove this relation using statistical physics. The significance of this result is that it allows us to understand the structure of the landscape using topology. I close by presenting some preliminary results regarding using topology to understand the optimization using machine learning. Here too, we use knot theory and topology to understand the structure of the energy landscape. We use embeddings of graphs to understand the relation between the architecture of a machine learning model and its efficiency in finding an optimal solution in the graph layout problem.
Thursday April 1, 2021 at 12:00 PM in Zoom