The task of estimating matrix based on its noisy, partial observation has emerged as the canonical challenge across a variety of fields spanning Inference, Machine Learning and Statistics
over the past decade or so.

Popularized examples abound, including Recommendation systems, Asymptotic Graph Theory (e.g. Graphons), Network Science (e.g. Community Detection), Social Data Processing (e.g. Ranking and Crowd Sourcing), Causal Inference (e.g. Synthetic Control), Panel Data Analysis, Bio-informatics (e.g. DNA sequencing) and more.

The purpose of this tutorial is to provide a comprehensive survey of various algorithmic and analytic approaches developed over the past decade across fields of information sciences, broadly defined. The goal is to ground these developments in the context of a “universal” model through connections with the theory of exchangeability (i.e. De Finetti (1937), Aldous and Hoover (1980s)). A particular attention will be paid to statistical and computational trade-off that arise in this class of problems. Open questions pertaining to conjectured fundamental limits and mysterious empirical algorithmic successes will be discussed.