Matrix Estimation Meets Statistical Network Analysis: Extracting low-dimensional structures in high dimension

The study of complex relationships among the elements of a large collection of random variables lead to the development of a number of areas in probability and statistics such as probabilistic network analysis or random matrix theory. The aim of the workshop was to address the challenge to develop a coherent mathematical framework within which these areas can be integrated, for a successful analysis of massive and complicated data sets. Mathematics Subject Classification (2010): 62G99, 60G05. Introduction by the Organisers The workshop Matrix estimation meets statistical network analysis, organized by Florentina Bunea (Ithaca), Angelika Rohde (Freiburg), Patrick Wolfe (London), and Harrison Zhou (New Haven) was well attended with around 25 participants with broad geographic representation. In summary, the workshop was devoted to the problem of developing a coherent mathematical framework within which the areas of probabilistic network analysis and random matrix theory can be integrated for a successful analysis of massive and complicated data sets. The workshop was a nice blend of researchers with the different backgrounds of matrix estimation and statistical network analysis. Having the two overlapping, but still different, communities together was crucial and we view the intensive interactions in the Oberwolfach setting as instrumental in making a synergy happen. Within lively discussions during and after the talks, and in particular in the evenings, fruitful inspiration for new research topics and directions has developed. Starting on 1746 Oberwolfach Report 29/2018 Monday morning with Boaz Nadler’s (Rechovot) first talk and the subsequent representation of Peter Bickel (Berkeley), a remarkably open atmosphere was already generated and remained present for the rest of the week. Whenever we observe entities and relations between them, either directly or induced from other data as a means of summarizing sparse dependency structure, we must draw inferences from network data. These datasets are growing so rapidly in complexity and dimensionality that our statistical analysis methods struggle to keep pace. For this reason network models have seen a revival in the last decade, one of the goals being to depart from basic random graph models such as those due to Erdős & Rényi (1959), which have limited applicability to genetics, neuroscience, internet networks, astronomy, and other fields where data exhibit a much more complicated dependence structure. Often networks come without labeled nodes, and the goal is to infer these labels (e.g., who belongs to which social group, or which proteins are implicated in what biological processes). At other times we must infer the network structure itself. More generally, as we encounter bigger and more heterogenous networks in the real world, understanding the limiting behavior of networks in the large-sample limit is critical to enabling statistical modeling and inference algorithms with good theoretical properties. An important problem is to develop network models that are highly heterogeneous even in the large-sample limit. By addressing this question, the workshop helped the community to take a large step forward. Another important statistical question is that of estimating the network. It is immediately clear that the term network is ambiguous and needs further clarification. Whereas in all cases, the pictorial representation is a graph consisting of nodes and edges between some of the nodes, the rule according to which an edge appears in such a graph is crucial in defining a certain type of network. For this reason, another core area of the workshop was the study of sparse matrix models and graphs in high dimension. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1641185, “US Junior Oberwolfach Fellows”. Matrix Estimation Meets Statistical Network Analysis 1747 Workshop: Matrix Estimation Meets Statistical Network Analysis: Extracting low-dimensional structures in high dimension