|Regularization is model styling|
First, we first look at data .
Then we think about a model ,
and an operator to link the model and the data.
Sometimes, the operator is merely the first term in a series expansion
Then, we fit
To fit the model, we must reduce the fitting residuals.
Realizing the importance of a data residual
is not always simply the size of the residual,
but is a function of it,
we conjure up (topic for later chapters)
a weighting function (which could be a filter) operator .
With we define our data residual:
Next, we realize that the data might not be adequate to determine the model,
perhaps because our comfortable dense sampling of the model
ill fits our economical sparse sampling of data.
Thus we adopt a fitting goal that mathematicians call ``regularization,''
and we might call a ``model styling'' goal
or more simply,
a quantification of our preconception of the best model.
We quantify our goals by choosing an operator ,
often simply a roughener like a gradient
(the choice again a topic in this and later chapters).
It defines our model residual by
, say we choose:
In an ideal world,
our model preconception (prejudice?)
would not conflict with measured data,
but real life is much more interesting than that.
The reason we pay for data acquisition
is that conflicts between data and preconceived notions invariably arise.
We need an adjustable parameter
that measures our ``bullheadedness,'' how much we intend
to stick to our preconceived notions in spite of contradicting data.
This parameter is generally called epsilon ,
because we like to imagine that our bullheadedness (prejudice?) is small.
(In mathematics, is often taken to be
an infinitesimally small quantity.)
Although any bullheadedness seems like a bad thing,
it must be admitted that measurements are imperfect too.
Thus, as a practical matter, we often find ourselves minimizing:
Although we often ignore in discussing the formulation of an application, when time comes to solve the problem, reality intercedes. Generally, has different physical units than (likewise and ), and we cannot allow our solution to depend on the accidental choice of units in which we express the problem. I have had much experience choosing , but it is only recently that I boiled it down to the suggestions of equations (22) and (25). Normally I also try other values, like double or half previous choices, and I examine the solutions for subjective appearance. The epsilon story continues in Chapter .
Computationally, we could choose a new with each iteration, but it is more expeditious to freeze , solve the problem, recompute , and solve the problem again. I have never seen a case in which more than one repetition was necessary.
People who work with small applications (less than about vector components) have access to an attractive theoretical approach called ``cross-validation.'' Simply speaking, we could solve the problem many times, each time omitting a different data value. Each solution would provide a model that could be used to predict the omitted data value. The quality of these predictions is a function of which provides a guide to finding it. My objections to cross validation are two-fold: First, I do not know how to apply it in the large applications we solve in this book (I should think more about it); and second, people who worry much about , perhaps first should think more carefully about their choice of the filters and , which is the focus of this book. Notice that both and can be defined with a scaling factor like . Often more important in practice, with and we have a scaling factor that need not be constant but can be a function of space or spatial frequency within the data space and/or model space.
|Regularization is model styling|