  Future large scale cosmological surveys will provide huge data sets whose
analysis requires efficient data compression. Calculating accurate covariances
is extremely challenging with increasing number of statistics used. Here we
introduce a formalism for achieving efficient data compression, based on a
local expansion of statistical measures around a fiducial cosmological model.
We specifically apply and test this approach for the case of cosmic shear
statistics. We demonstrate the performance of our approach, using a Fisher
analysis on cosmic shear tomography described in terms of E-/B-mode separating
statistics (COSEBIs). We show that our data compression is highly effective in
extracting essentially the full cosmological information from a strongly
reduced number of observables. Specifically, the number of statistics needed
decreases by at least one order of magnitude relative to the COSEBIs, which
already compress the data substantially compared to the shear two-point
correlation functions. The efficiency appears to be affected only slightly if a
highly inaccurate covariance is used for defining the compressed statistics,
showing the robustness of the method. We conclude that an efficient data
compression is achievable and that the number of compressed statistics depends
on the number of model parameters.
  In addition, we study how well band powers can be obtained from measuring
shear correlation functions over a finite interval of separations. We show the
strong limitations on the possibility to construct top-hat filters in Fourier
space, for which the real-space analog has a finite support, yielding strong
bounds on the accuracy of band power estimates. The error on an estimated
band-power is larger for a narrower filter and a smaller angular range which
for relevant cases can be as large as 10%.
