  Covariance matrices are among the most difficult pieces of end-to-end
cosmological analyses. In principle, for two-point functions, each component
involves a four-point function, and the resulting covariance often has hundreds
of thousands of elements. We investigate various compression mechanisms capable
of vastly reducing the size of the covariance matrix in the context of cosmic
shear statistics. This helps identify which of its parts are most crucial to
parameter estimation. We start with simple compression methods, by isolating
and "removing" 200 modes associated with the lowest eigenvalues, then those
with the lowest signal-to-noise ratio, before moving on to more sophisticated
schemes like compression at the tomographic level and, finally, with the
Massively Optimized Parameter Estimation and Data compression (MOPED). We find
that, while most of these approaches prove useful for a few parameters of
interest, like $\Omega_m$, the simplest yield a loss of constraining power on
the intrinsic alignment (IA) parameters as well as $S_8$. For the case
considered -- cosmic shear from the first year of data from the Dark Energy
Survey -- only MOPED was able to replicate the original constraints in the
16-parameter space. Finally, we apply a tolerance test to the elements of the
compressed covariance matrix obtained with MOPED and confirm that the IA
parameter $A_{\mathrm{IA}}$ is the most susceptible to inaccuracies in the
covariance matrix.
