Visualization of High-dimensional Scalar Functions Using Principal Parameterizations

TL;DR

This paper introduces a principal component-based visualization method for high-dimensional scalar fields that captures parameter sensitivities and provides geometrical insights, using tensor decomposition for practical implementation.

Contribution

It presents a novel dimensionality reduction approach in the $L^2$ space for visualizing high-dimensional scalar functions, linking it with sensitivity analysis and enabling interactive visualization.

Findings

Method accurately reflects parameter sensitivities.

Provides geometrical interpretation of variance-based sensitivity.

Enables interactive visualization of complex models.

Abstract

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.