A Bayesian Framework for Regularization

Regularization is a popular method for interpolating sparse data, as well as smoothing data obtained from noisy measurements. Simply put, regularization looks for an interpolating or approximating function which is both close to the data and also "smooth" in some sense. Formally, this function is obtained by minimizing an error functional which is the sum of two terms, one measuring the distance from the data, the other measuring the smoothness of the function. The classical approach to regularization is: decide the relative weights that should be given to these two terms, and minimize the resulting error functional. This approach, however, suffers from two serious flaws: there is no rigorous way to compute these weights and it does not find the function which is the MAP estimate for the interpolation problem. This may result in grave instability in the reconstruction process; two data sets which are arbitrarily close to each other can be assigned two radically different interpolants. ..