  We apply the projective truncation technique to the tensor renormalization
group (TRG) algorithm in order to reduce the computational cost from
$O(\chi^6)$ to $O(\chi^5)$, where $\chi$ is the bond dimension, and propose
three kinds of algorithms for demonstration. On the other hand, the technique
causes a systematic error due to the incompleteness of a projector composed of
isometries, and in addition requires iteration steps to determine the
isometries. Nevertheless, we find that the accuracy of the free energy for the
Ising model on a square lattice is recovered to the level of TRG with a few
iteration steps even at the critical temperature for $\chi$ = 32, 48, and 64.
