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SUMMARY:Optimal approximation of infinite-dimensional\, Banach-valued\, ho
 lomorphic functions from i.i.d. samples - Ben Adcock (Simon Fraser Univers
 ity)
DTSTART:20240717T083000Z
DTEND:20240717T091000Z
UID:TALK218161@talks.cam.ac.uk
DESCRIPTION:The problem of approximating Banach-valued functions of infini
 tely-many variables has been studied intensively over the last decade\, du
 e to its application in\, for instance\, computational uncertainty quantif
 ication. Here\, functions of this type arise as solution maps of various p
 arametric or stochastic PDEs. In this talk\, I will discuss recent work on
  the approximation of such functions from finite samples. First\, I will d
 escribe new lower bounds for various (adaptive) m-widths of classes of suc
 h functions. These bounds show that any combination of (adaptive) linear s
 amples and a (linear or nonlinear) recovery procedure can at best achieve 
 certain algebraic of convergence with respect to the number of samples. Ne
 xt\, I will focus on the case where the samples are i.i.d. pointwise sampl
 es from some underlying probability measure\, as is commonly encountered i
 n practice. I will discuss methods that construct multivariate polynomial 
 approximations via least squares and compressed sensing. As I will show\, 
 these methods attain matching upper bounds\, up to polylogarithmic factors
 . In particular\, this implies that i.i.d. pointwise samples constitute ne
 ar-optimal information for this problem and these schemes constitute near-
 optimal methods for reconstruction from such data. Finally\, time permitti
 ng\, I will discuss how these results can be extended to the problem of op
 erator learning\, yielding near-optimal guarantees for learning classes of
  holomorphic operators related to parametric PDEs.\nCo-authors: Simone Bru
 giapaglia (Concordia)\, Nick Dexter (Florida State University) and Sebasti
 an Moraga (Simon Fraser University)
LOCATION:Seminar Room 1\, Newton Institute
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