How do you define the structure of a Covariance Matrix

Question

I am reading a paper by Ledoit and Wolf,2001 on Improved Estimation of the Covariance Matrix of Stock Returns. I am a little confused by some points, and will appreciate a broader explanation.

The paper is attached here

Specifically, a paragraph states:

The traditional estimator — the sample covariance matrix — is seldom used because it imposes too little structure.

Then it proceeds to say

The cure is to impose some structure on the estimator. Ideally, the particular form of the structure should be dictated by the problem at hand. In the case of stock returns, a low-dimensional factor structure seems natural.

Then another paragraph states

One possible way is to specify a K-factor model with uncorrelated residuals. Then K controls how much structure we impose: the fewer the factors, the stronger the structure

My questions are the following:

How do you define the structure of the covariance matrix? How does one measure how good this structure is?

Answer 1

This other answer here explains why is it problematic the estimation of the covariance of the matrix, in the same sense as they mention at the beginning of the paper.

They talk in terms of structure to say that you have no assumptions on your data. What they do is to impose some regularization, so that the resulting covariance matrix is not singular (is well conditioned).