2:00pm, Thursday, 20 September 2007:
<br>
<br>
<center><b>
Regressing without supervision
</b>
<br>
<br> 
Martin Raphan
<br>
<br>
</center>

There are two standard frameworks for describing optimal estimation of a
random quantity from corrupted measurements. The first technique uses
explicit models of both the corruption process and the prior
distribution of the quantity to be estimated in order to formulate an
optimal estimator via Bayes' rule. The second technique uses supervised
training on a data set, which has clean samples paired with corrupted
versions of those samples, to choose an optimal estimator from some family.
In many applications, however, one has available neither a model of the
prior distribution, nor uncorrupted measurements of the variable being
estimated. We will describe a linear algebraic framework for expressing
the least squares estimator (regression function) entirely in terms of a
model of the corruption process and the density of the corrupted
measurements. We show practical implementations of these nonparametric
estimators for various corruption models, and demonstrate the use of this
procedure for denoising photographic images corrupted by additive Gaussian
noise. We also describe a dual, prior-free formulation of the Mean Square
Error (MSE) and show how this may be used for the selection of optimal
estimators from a parametric family. We then demonstrate the use of this dual
formulation in image denoising.