  3D data compression techniques can be used to determine the natural basis of
radial eigenmodes that encode the maximum amount of information in a
tomographic large-scale structure survey. We explore the potential of the
Karhunen-Lo\`eve decomposition in reducing the dimensionality of the data
vector for cosmic shear measurements, and apply it to the final data from the
\cfh survey. We find that practically all of the cosmological information can
be encoded in one single radial eigenmode, from which we are able to reproduce
compatible constraints with those found in the fiducial tomographic analysis
(done with 7 redshift bins) with a factor of ~30 fewer datapoints. This
simplifies the problem of computing the two-point function covariance matrix
from mock catalogues by the same factor, or by a factor of ~800 for an
analytical covariance. The resulting set of radial eigenfunctions is close to
ell-independent, and therefore they can be used as redshift-dependent galaxy
weights. This simplifies the application of the Karhunen-Lo\`eve decomposition
to real-space and Fourier-space data, and allows one to explore the effective
radial window function of the principal eigenmodes as well as the associated
shear maps in order to identify potential systematics. We also apply the method
to extended parameter spaces and verify that additional information may be
gained by including a second mode to break parameter degeneracies. The data and
analysis code are publicly available at
https://github.com/emiliobellini/kl_sample.
