# Active Manifolds: A non-linear analogue to Active Subspaces

arXiv: https://arxiv.org/abs/1904.13386 · HTML: https://mlsanity.com/paper/1904.13386/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2019.

## Built on by (most cited first, 8 of 8)

- [A survey and benchmark of high-dimensional Bayesian optimization of discrete sequences](https://mlsanity.com/paper/2406.04739/) (2024)
- [Sensitivity Prewarping for Local Surrogate Modeling](https://mlsanity.com/paper/2101.06296/) (2021)
- [Neural active manifolds: nonlinear dimensionality reduction for uncertainty quantification](https://mlsanity.com/paper/2408.03534/) (2024)
- [Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations](https://mlsanity.com/paper/2104.14072/) (2021)
- [Surrogate to Poincar'e inequalities on manifolds for dimension reduction in nonlinear feature spaces](https://mlsanity.com/paper/2505.01807/) (2025)
- [A local approach to parameter space reduction for regression and classification tasks](https://mlsanity.com/paper/2107.10867/) (2021)
- [Conditional Korhunen-Lo'eve regression model with Basis Adaptation for high-dimensional problems: uncertainty quantification and inverse modeling](https://mlsanity.com/paper/2307.02572/) (2023)
- [A Riemannian View on Active Subspaces](https://mlsanity.com/paper/2607.25163/) (2026)

## Abstract (arXiv)

We present an approach to analyze C1(ℝm) 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.