  We apply two compression methods to the galaxy power spectrum
monopole/quadrupole and bispectrum monopole measurements from the BOSS DR12
CMASS sample. Both methods reduce the dimension of the original data-vector to
the number of cosmological parameters considered, using the Karhunen-Lo\`eve
algorithm with an analytic covariance model. In the first case, we infer the
posterior through MCMC sampling from the likelihood of the compressed
data-vector (MC-KL). The second, faster option, works by first Gaussianising
and then orthogonalising the parameter space before the compression; in this
option (G-PCA) we only need to run a low-resolution preliminary MCMC sample for
the Gaussianization to compute our posterior. Both compression methods
accurately reproduce the posterior distributions obtained by standard MCMC
sampling on the CMASS dataset for a $k$-space range of
$0.03-0.12\,h/\mathrm{Mpc}$. The compression enables us to increase the number
of bispectrum measurements by a factor of $\sim 23$ over the standard binning
(from 116 to 2734 triangles used), which is otherwise limited by the number of
mock catalogues available. This reduces the $68\%$ credible intervals for the
parameters $\left(b_1,b_2,f,\sigma_8\right)$ by
$\left(-24.8\%,-52.8\%,-26.4\%,-21\%\right)$, respectively. The best-fit values
we obtain are $(b_1=2.31\pm0.17,b_2=0.77\pm0.19,$
$f(z_{\mathrm{CMASS}})=0.67\pm0.06,\sigma_8(z_{\mathrm{CMASS}})=0.51\pm0.03)$.
Using these methods for future redshift surveys like DESI, Euclid and PFS will
drastically reduce the number of simulations needed to compute accurate
covariance matrices and will facilitate tighter constraints on cosmological
parameters.
