This paper considers a way of generalizing the t-SVD third-order tensors (regarded as tubal matrices) to arbitrary order $$N$$ [which can be similarly regarded $$(N-1)$$ ]. Such generalization is different from for greater than three (Martin et al. in SIAM J Sci Comput 35(1):A474–A490, 2013). The decomposition called Hot-SVD, since it recognized tensor–tensor product version celebrated higher S...