  The higher-order tensor renormalization group (HOTRG) is a fundamental method
to calculate the physical quantities by using a tensor network representation.
This method is based on the singular value decomposition (SVD) to take the
contraction of all indices in the network with an approximation. For the SVD,
randomized singular value decomposition (R-SVD) is a powerful method to reduce
computational costs of SVD. However, HOTRG with the randomized method is not
established. We propose a randomized HOTRG method in a dimension $d$ with the
computational cost $O(D^{3d})$ depending on the truncated bond dimension $D$.
We also introduce the minimally-decomposed TRG (MDTRG) as the R-HOTRG on the
tensor of order $d+1$ with $O(D^{2d + 1})$ and a triad representation of the
MDTRG (Triad-MDTRG) with $O(D^{d+3})$. The results from these formulations are
consistent with the HOTRG result with the same truncated bond dimension $D$.
