A new version of High Dimensional Model Representation (HDMR) is presented in this work. Vector HDMR has been quite recently developed to deal with the decomposition of vector valued multivariate functions. It was an extension from scalars to vectors by possibly using matrix weights. However, that expansion is based on an ascending multivariance starting from a constant term via a set of appropriately imposed conditions which can be related to orthogonality in a conveniently chosen Hilbert space. This work adds more flexibility by introducing certain matrix valued univariate support functions. We assume weight matrices proportional to unit matrices. This work covers only the basic issues related to the fundamental elements of the new approach.