EMPTY BINS AND PRECONDITIONING

There are at least three ways to fill empty bins. They seem to be all equivalent, though that is not as obvious as I would like it to be.

The original way in Chapter is to restore missing data by ensuring that the restored data, after specified filtering, has minimum energy, say .Introduce the selection mask operator , a diagonal matrix with ones on the known data and zeros elsewhere (on the missing data). Thus or   where we have defined to be the data with missing values set to zero by .

A second way to find missing data is with the set of goals   and take the limit as the scalar .At that limit, we should have the same result as equation (27).

A third way to find missing data is to precondition equation (28), namely, try the substitution $\bold m = \bold A^{-1} \bold p$.  I think (hope) it is proven later that if we start from and if we are interested in the limit we can simply forget about the fitting goal .