Active Manifolds: A non-linear analogue to Active Subspaces
This provides a more accurate method for sensitivity analysis and approximation in high-dimensional computational models, though it is computationally more expensive.
The authors tackled limitations of the Active Subspaces method by proposing Active Manifolds, a non-linear technique that identifies a 1-D curve in high-dimensional domains for function approximation and sensitivity analysis. They demonstrated that Active Manifolds reduces approximation error by an order of magnitude compared to Active Subspaces and provides more detailed parameter sensitivity information in a magnetohydrodynamic model.
We present an approach to analyze $C^1(\mathbb{R}^m)$ functions that addresses limitations present in the Active Subspaces (AS) method of Constantine et al.(2015; 2014). Under appropriate hypotheses, our Active Manifolds (AM) method identifies a 1-D curve in the domain (the active manifold) on which nearly all values of the unknown function are attained, and which can be exploited for approximation or analysis, especially when $m$ is large (high-dimensional input space). We provide theorems justifying our AM technique and an algorithm permitting functional approximation and sensitivity analysis. Using accessible, low-dimensional functions as initial examples, we show AM reduces approximation error by an order of magnitude compared to AS, at the expense of more computation. Following this, we revisit the sensitivity analysis by Glaws et al. (2017), who apply AS to analyze a magnetohydrodynamic power generator model, and compare the performance of AM on the same data. Our analysis provides detailed information not captured by AS, exhibiting the influence of each parameter individually along an active manifold. Overall, AM represents a novel technique for analyzing functional models with benefits including: reducing $m$-dimensional analysis to a 1-D analogue, permitting more accurate regression than AS (at more computational expense), enabling more informative sensitivity analysis, and granting accessible visualizations(2-D plots) of parameter sensitivity along the AM.