Fast orthogonal decomposition of Volterra cubic kernels using oblique unfolding

Discrete-time Volterra modeling is a central topic in many application areas and a large class of nonlinear systems can be modeled using high-order Volterra series. The problem with Volterra series is that the number of parameters grows very rapidly with the order of the nonlinearity and the memory in the system. In order to efficiently implement this model, kernel eigen-decomposition can be used in the context of a Parallel-Cascade realization of a Volterra system. So, using the multilinear SVD (HOSVD) for decomposing high-order Volterra kernels seems natural. In this paper, we propose to drastically reduce the computational cost of the HOSVD by (1) considering the symmetrized Volterra kernel and (2) exploiting the column-redundancy of the associated mode by using an oblique unfolding of the Volterra kernel. Keeping in mind that the complexity of the full HOSVD for a cubic (I x I x I) unstructured Volterra kernel needs 12I⁴ flops, our solution allows reducing the complexity to 2I⁴ flops, which leads to a gain equal to six for a sufficiently large size I.