In many situations, given a sequence of $n$ random variables (or sensors) 
the covariance matrix is not know but has to be estimated. In the case we 
have $m$ observations and $m>>n$ then the sample covariance matrix is a good 
approximation of the true covariance. More specifically, for a fixed number 
of variables the sample covariance matrix converges with $m$ to the true 
covariance matrix. However, in applications like weather forecast, wireless 
communications (MIMO channels with a big number of antennas), linear estimation 
and military applications the number of observations is limited and usually one 
has $m\leq n$. The traditional remedy is diagonal loading - the addition of a 
small identity matrix to make the covariance estimate invertible. In this talk 
we will discuss an alternative approach. In this one, we reduce the dimension 
of the data through an ensemble of isotropically random (Haar measure) unitary 
matrices. For every member of the unitary ensemble, the shortened data vectors 
yield a statistically meaningful, invertible covariance estimate from which we 
can compute an estimate for the ultimate desired quantity. Finally, we take the 
expectation of this estimate with respect to the unitary ensemble. Preliminary 
numerical results indicate considerable promise for this approach. This is 
based on a project done at Bell Labs with Steven H. Simon and Gabriel H. Tucci 
where we used techniques from random matrices, free probability and representation
theory.