The following was written by Heather A. Harrington, Hooke Research Fellow at the Mathematical Institute at the University of Oxford.
The SAMSI LDHD workshop covered a range of topics with the overarching theme of using mathematical and statistical techniques to analyze high dimensional systems/data. Many of the talks focused on exploiting features of such systems (e.g., sparity, structure etc.) to compute high dimensional covariances, perform convex optimization, classify images, and solve inverse and inference problems using Bayesian statistics and graphical models. I came to the conference very open and interested in many aspects of capturing the low dimensional features of high dimensional data and I learned the many advances in approaches as well as the current challenges in the field.
I particularly found the talks on computational geometry and topology fascinating. Recently I have became interested in persistent homology for analyzing networks, but I hadn’t considered some of the open problems on the statistical side. I found the most interesting talks during the LDHD conference were by Sayan Mukherjee on sufficient statistics for shapes and graph Laplacians, Ingrid Daubechies on conformal mapping to define a metric between surfaces, as well as David Dunson’s introduction on Bayesian inference, manifold learning, and geometric type models for exploiting the low dimensional structure in high dimensional data.
Overall, the workshop provided a broad overview of the field, but more specifically, I learned about current research in combining probabilistic and statistical models for geometric/topological structures, and I’ve now joined two working groups aiming to understand how to put a likelihood function on surfaces and/or the meaning of noise in computational topology diagrams (summaries).