Instead of using a sample covariance matrix for portfolio optimization, Ledoit and Wolf use an estimator that is the weighted average of the sample covariance matrix and the identity matrix, $I$. This approach can be interpreted as a method that shrinks the sample covariance matrix toward the identity matrix, pulling the most extreme coefficients toward more central values, systematically reducing estimation error when it matters most.
The identity matrix contains 0's for off-diagonals, and 1's for the diagonal entries. Is the essence of the importance of the identity matrix in portfolio theory due to the fact that $I$ represents a noiseless data structure due to its off-diagonals being 0? Or, instead of noise, does its supposed ideal properties come more from the concept of sparsity?
If so, does this mean that any covariance matrix whose off-diagonals are much smaller than its diagonals must therefore be more amenable to invertibility and quadratic optimization with low estimation error? Or what exactly is so great about a symmetric matrix whose off-diagonals are much smaller than the diagonal elements?