Isotonic Regression in General Dimensions – Richard Samworth, University of Cambridge

Explore isotonic regression in multiple dimensions through this 48-minute lecture by Richard Samworth from the University of Cambridge. Delve into the challenges of reconstructing complex processes with numerous parameters, addressing the curse of dimensionality in high-dimensional function approximation. Learn about modern approaches that overcome limitations by employing structural assumptions like low intrinsic dimensionality and sparse representations. Examine topics such as multidimensional covariance, natural partial ordering, and Dijkstra's algorithm. Investigate least squares estimators, statistical dimensions, and global empirical risk minimization. Gain insights into adaptation, lower bounds, monotonic functions, and random design results. Conclude with a discussion of the adage carry theorem and a comprehensive summary of isotonic regression in general dimensions.