I'm trying to compare the efficiency of different estimators of the covariance matrix of a particular type of multivariate normally distributed data. This comparison, as well as the estimation process itself, requires the computation of a loss function that gives a metric on the difference between the estimated matrix and some reference (e.g. the sample covariance matrix).
My problem is that all the most commonly used loss functions for this problem - or at least those that I've been able to find - include the inverse of the covariance estimate as one of the terms. My dataset happens to have more features than observations, so the estimated covariance matrix is always singular, which means I can't compute any of these loss functions.
So my question: what sort of loss function would be appropriate in this scenario? Ideally I would like to know if there is an accepted, standard choice for this, but any suggestions would be very much appreciated!