Final Report Summary - AGALT (Asymptotic Geometric Analysis and Learning Theory)
The main aim of this project was the study of the interplay between Asymptotic Geometric Analysis, an area focusing on the geometry of convex, centrally-symmetric bodies in , and Learning Theory, in which one investigates the difficultly of prediction problems.
The results we obtained showed that indeed, certain probabilistic phenomena, including the difficulty of statistical prediction problems, are exhibited by the geometry of certain random sets, naturally associated with the underlying class. The geometry of these sets leads to an accurate description of the behaviour of certain empirical processes and to a complete characterization of the error rate in many prediction problems.
This approach has led to a solution of several fundamental questions in Statistics (e.g. sharp estimates on the error rate of several learning algorithm, including the empirical risk minimization algorithm), Probability (estimates on the largest and smallest singular value of a random matrix with iid rows) and Geometry (structural results on random, non-gaussian, projections of convex bodies).
The results we obtained showed that indeed, certain probabilistic phenomena, including the difficulty of statistical prediction problems, are exhibited by the geometry of certain random sets, naturally associated with the underlying class. The geometry of these sets leads to an accurate description of the behaviour of certain empirical processes and to a complete characterization of the error rate in many prediction problems.
This approach has led to a solution of several fundamental questions in Statistics (e.g. sharp estimates on the error rate of several learning algorithm, including the empirical risk minimization algorithm), Probability (estimates on the largest and smallest singular value of a random matrix with iid rows) and Geometry (structural results on random, non-gaussian, projections of convex bodies).