Optimal Representation and Visualization of Multivariate Data and Functions in Low-Dimensional Spaces

Abstract

The analysis and processing of massive amount of multivariate data and high-dimensional functions have become a basic need in many areas of science and engineering. To reduce the dimensionality for compact representation and visualization of high-dimensional information appear imperative in exploratory research and engineering modeling. Since D. Hilbert raised the 13thproblem in 1900, the study on possibility to express high-dimensional functions via composition of lower-dimensional functions has gained considerable success[1, 2]. Nonetheless, no methods of realization are ever indicated, and not even all integrable functions can be treated this way, a fortiori functions in L2(Ω). The common practice is to expand high-dimensional functions into a convergent series in terms of a chosen orthonormal basis with lower dimensional ones. However, the length and rapidity of convergence of the expansion heavily depend upon the choice of basis. In this paper we briefly report some new results of study in seeking an optimal basis for a given function provided with fewest terms and rapidest convergence. All elements of the optimal basis turned out to be products of single-variable functions taken from the unit balls of ingredient spaces. The proposed theorems and schemes may find wide applications in data processing, visualization, computing, engineering simulation and decoupling of nonlinear control systems. The facts established in the theorems may have their own theoretical interests.