On the interface between Numerical Approximation, Inference and Learning

Although numerical approximation, statistical inference and learning are traditionally seen as entirely separate subjects, they are intimately connected through the common purpose of making estimations with partial information. This talk is an invitation to explore these connections from the consolidating perspective of game/decision theory and it is motivated by the suggestion that these confluences might not just be objects of curiosity but can constitute a pathway to simple solutions to fundamental problems in all three areas. We will illustrate this point through problems related to numerical homogenization, operator adapted wavelets, computation with dense kernel matrices and to the kernel selection/design problem in Machine Learning. In these interplays, accurate reduced/multiscale models (for PDEs) can be identified as optimal bets for adversarial games describing the process of computing with partial information. Moreover, efficient kernels (for ML) can be selected by using relative energy content at fine scales (with a notion of scale corresponding to the number of data points) as an ordering criterion leading to the identification of (data driven) flows in kernel spaces (Kernel Flows), that (1) enable the design of bottomless networks amenable to some degree of analysis (2) appear to converge towards kernels with good generalization properties.

This talk will cover joint work with F. Schäfer, C. Scovel, T. Sullivan, G. R. Yoo and L. Zhang.