Weekly Bulletin

The statistical behavior of a random tensor is controlled by its moments. Unfortunately, these moments are notoriously challenging to describe analytically. In this talk, I will present the Penrose diagrammatic notation as a visual calculus that organizes these calculations into a series of diagrams. As an example, we will investigate the fourth-moment behavior of a particular random tensor called a random matrix product state (rMPS), also known as a random tensor train. Using a percolation argument, we will characterize the regime where an rMPS agrees with a standard Gaussian random tensor up to the first four moments. No prior background in tensors is assumed. Joint work with Joel A. Tropp, Ethan Epperly, and Raphael Meyer.

Cancelled: The advent of drones with computing units provides new opportunities and challenges for statistical sampling. Especially so in the ocean where remote communication and control is difficult. In this presentation, we focus on marine robots and in particular so-called autonomous underwater vehicles (AUV) that can be deployed for in-situ ocean sampling useful for surveying, monitoring or mapping purposes. Equipped with a computer unit, an AUV can sense the ocean environment, update its onboard model based on data, plan where and when to navigate for efficient spatio-temporal sampling efforts, and act using its engines and navigation systems. We demonstrate methods for combining spatio-temporal statistical modeling and autonomous robotic systems for efficient experimental designs. Building on Gaussian random field models, we present an approach for real-time sequential AUV operation and planning. Depending on the goal of the operation, this adaptive sampling approach relies on fast computation of acquisition functions like maximum expected improvement, minimum expected Bernoulli variance or entropy which have closed-form solutions for the Gaussian model and hence enable real-time computation for efficient sampling. In several applications with ocean front mapping, mine tailings pollution monitoring and chlorophyll hotspot sampling, we develop, test and deploy algorithms for efficient AUV sampling.