Randomized higher-order tensor renormalization group for higher dimension

We introduce the new formulation of the tensor renormalization group(TRG) in the higher dimension 
with the randomized singular value decomposition(SVD). The randomized SVD is one popular method 
to reduce the computational cost of SVD, and is also applied to the reduction of the contraction of the tensors. 
The randomized SVD can reduce the cost. We also introduce the minimally-decomposed TRG (MDTRG) and its triad representation, and find the precision is comparable to the original higher-order TRG without additional systematic error. 
Our formulations include general idea for the TRG, and could be one good choice to calculate the physical quantities such as the partition function without sign-problem.