I have a correlation matrix $A$ for an equity market that is not positive definite. Higham (2002) proposes the Alternating Projections Method, minimising the weighted Frobenius norm $||A-X||_W$ where $X$ is the resulting positive definite matrix.
How should one choose the weight matrix $W$?
The easy alternative is to weigh them equally (W is an identity matrix), but if one has exposures to a portfolio, wouldn't it be natural to weigh the correlations according to your weights of exposure in the different assets, in order to alter their historical correlation less than for those assets you have little exposure in? Or is there a more natural choice?