**Gilles Blanchard**

Wednesday 4th November 2015

**Change in Time: 3.00pm**

** **

Ground Floor Seminar Room

25 Howland Street, London, W1T 4JG

__Convergence rates of for spectral regularization methods for statistical inverse learning problems__

Consider an inverse problem of the form g = Af , where A is a known

operator between Hilbert function spaces, and assume that we observe g

at some
randomly drawn points X_1,...,X_n which are i.i.d. according to some

distribution P_X, and where additionally each observation is subject

to a random independent noise. The goal is to recover the function

g. Here it is assumed that for each point x the evaluation mapping

f -> Af (x) is continuous. This setting as well as its relation to

random nonparametric regression and statistical learning with

reproducing kernels has been proposed and studied in particular in a

series of works
by Caponnetto, De Vito, Rosasco, and Odone (between others). In this

talk we will
first review this setting in some detail, as well as the principle

of estimation by so-called spectral methods. We will present some results

concerning convergence rates of such methods that extend and complete

previously known ones. In particular, we will consider the optimality,

from a statistical
point of view, of a general class of linear spectral methods.