An efficient method was recently introduced [J. Chem. Phys. 102, 5605 (1995); 104, 7974 (1996)] to represent multidimensional potential energy surfaces as a linear combination of products of one-dimensional functions, so-called natural potentials. Weight functions were shown to be easily implemented in the product representation scheme as long as they are separable, i.e., defined as a product of one-dimensional weight functions. Here the constraint imposed by the special product form of the separable weights is removed. Nonseparable weights are emulated by dividing the potential energy surface in arbitrary regions of minor and major physical relevance and by utilizing a so-called relevant region iteration procedure. Maintaining the advantageous computational scaling properties of the product representation scheme, this relevant region iteration procedure allows the stepwise improvement of the surface representation in the regions of major relevance. The quality of the product representation in the regions of minor relevance remains nevertheless acceptable. As a consequence, the number of potential expansion coefficients can be reduced substantially. The product representation of potential energy surfaces is especially well suited to be employed within the framework of the multiconfiguration time-dependent Hartree (MCTDH) approximation. To check the performance of the proposed method the Liu-Siegbahn-Truhlar-Horowitz (LSTH) surface is represented in Jacobian coordinates, and initial-state selected reaction probabilities for the H+H-2(nu=j=0)-->H-2+H exchange reaction are computed.