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 $left(b_1,b_2,f,sigma_8right)$ by $left(-24.8%,-52.8%,-26.4%,-21%right)$, respectively. The best-fit values we obtain are $(b_1=2.31pm0.17,b_2=0.77pm0.19,$ $f(z_{mathrm{CMASS}})=0.67pm0.06,sigma_8(z_{mathrm{CMASS}})=0.51pm0.03)$. 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.
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