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.

Intro
Isotonic regression
Multidimensional covariance
Natural partial ordering
Independent observations
Dijkstras algorithm
Work on isotonic regression
Fixed lattice design
Least squares estimator
Bound the risk
Statistical dimension
Antichain
Global empirical risk minimization
Adaptation
Lower bound
Monotonic functions
Random design
Random design results
The adage carry theorem
Summary