Efficient Implementation of High Dimensional Model Representations

Physical models of various phenomena are often represented by a mathematical model where the output(s) of interest have a multivariate dependence on the inputs. Frequently, the underlying laws governing this dependence are not known and one has to interpolate the mathematical model from a finite number of output samples. Multivariate approximation is normally viewed as suffering from the curse of dimensionality as the number of sample points needed to learn the function to a sufficient accuracy increases exponentially with the dimensionality of the function. However, the outputs of most physical systems are mathematically well behaved and the scarcity of the data is usually compensated for by additional assumptions on the function (i.e., imposition of smoothness conditions or confinement to a specific function space). High dimensional model representations (HDMR) are a particular family of representations where each term in the representation reflects the individual or cooperative contributions of the inputs upon the output. The main assumption of this paper is that for most well defined physical systems the output can be approximated by the sum of these hierarchical functions whose dimensionality is much smaller than the dimensionality of the output. This ansatz can dramatically reduce the sampling effort in representing the multivariate function. HDMR has a variety of applications where an efficient representation of multivariate functions arise with scarce data. The formulation of HDMR in this paper assumes that the data is randomly scattered throughout the domain of the output. Under these conditions and the assumptions underlying the HDMR it is argued that the number of samples needed for representation to a given tolerance is invariant to the dimensionality of the function, thereby providing for a very efficient means to perform high dimensional interpolation. Selected applications of HDMR's are presented from sensitivity analysis and time-series analysis.