Visualization of High-dimensional Scalar Functions Using Principal Parameterizations
This work addresses the challenge of visualizing parameter spaces in computational science and engineering, offering a method that enhances interpretability of sensitivity analysis, though it appears incremental in its application of existing techniques.
The paper tackles the problem of visualizing high-dimensional scalar functions by proposing a principal component-based approach that maps partial functions to low-dimensional manifolds, directly linking visualization to Sobol's sensitivity analysis and enabling interactive analysis via tensor decomposition.
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.