Dimension Reduction Using Active Manifolds

TL;DR

This paper introduces Active Manifolds, a nonlinear dimension reduction method inspired by Active Subspaces, which improves the approximation of high-dimensional functions by seeking nonlinear manifolds, enhancing accuracy and interpretability.

Contribution

The paper develops a novel algorithm for nonlinear dimension reduction called Active Manifolds, extending the Active Subspaces approach to better approximate high-dimensional functions.

Findings

Increased approximation accuracy over linear methods

Guaranteed accessible visualization of the reduced space

Improved estimation accuracy for high-dimensional models

Abstract

Scientists and engineers rely on accurate mathematical models to quantify the objects of their studies, which are often high-dimensional. Unfortunately, high-dimensional models are inherently difficult, i.e. when observations are sparse or expensive to determine. One way to address this problem is to approximate the original model with fewer input dimensions. Our project goal was to recover a function f that takes n inputs and returns one output, where n is potentially large. For any given n-tuple, we assume that we can observe a sample of the gradient and output of the function but it is computationally expensive to do so. This project was inspired by an approach known as Active Subspaces, which works by linearly projecting to a linear subspace where the function changes most on average. Our research gives mathematical developments informing a novel algorithm for this problem. Our approach, Active Manifolds, increases accuracy by seeking nonlinear analogues that approximate the function. The benefits of our approach are eliminated unprincipled parameter, choices, guaranteed accessible visualization, and improved estimation accuracy.