  We propose a modified form of a tensor renormalization group algorithm for
evaluating partition functions of classical statistical mechanical models on 2D
lattices. This algorithm coarse-grains only the rows and columns of the lattice
adjacent to a single core tensor at each step, such that the lattice size
shrinks linearly with the number of coarse-graining steps as opposed to
shrinking exponentially as in the usual tensor renormalization group (TRG).
However, the cost of this new approach only scales as $O(\chi^4)$ in terms of
the bond dimension $\chi$, significantly cheaper than the $O(\chi^6)$ cost
scaling of TRG, whereas numerical benchmarking indicates that both approaches
have comparable accuracy for the same bond dimension $\chi$. It follows that
the new algorithm can allow for more accurate investigation of classical models
on modestly sized lattices than is possible with standard TRG.
