Visualization of High-dimensional Scalar Functions Using Principal Parameterizations

Insightful visualization of multidimensional scalar fields, in particular parameter spaces, is key to many fields in computational science and engineering. We propose a principal component-based approach to visualize such fields that accurately reflects their sensitivity to input parameters. The method performs dimensionality reduction on the vast $L^2$ Hilbert space formed by all possible partial functions (i.e., those defined by fixing one or more input parameters to specific values), which are projected to low-dimensional parameterized manifolds such as 3D curves, surfaces, and ensembles thereof. Our mapping provides a direct geometrical and visual interpretation in terms of Sobol's celebrated method for variance-based sensitivity analysis. We furthermore contribute a practical realization of the proposed method by means of tensor decomposition, which enables accurate yet interactive integration and multilinear principal component analysis of high-dimensional models.