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Active Subspaces in Infinite Dimension

Poorbita Kundu, Nathan Wycoff

TL;DR

The paper addresses dimension reduction for functionals in infinite dimensions by extending the Active Subspace concept to Hilbert spaces. It defines the active-subspace operator $\mathcal{C} = \mathbb{E}[\nabla f(U) \otimes \nabla f(U)]$, proves its spectral properties (compact, self-adjoint, trace-class), and derives the leading subspace $\mathcal{A}_n$ that captures the dominant gradient energy. A computable Monte Carlo estimator $\widehat{\mathcal{C}}_B$ is developed, with proven consistency, eigenvalue/eigenfunction convergence, a $O(B^{-1/2})$ rate under fourth-moment assumptions, and a Hilbert-space CLT, enabling reliable uncertainty quantification. Applications to PDE-constrained problems demonstrate how the resulting low-dimensional structure supports visualization, surrogate modeling, and Bayesian optimization in function spaces, with an open-source FEniCS-based implementation illustrating practical impact.

Abstract

Active subspace analysis uses the leading eigenspace of the gradient's second moment to conduct supervised dimension reduction. In this article, we extend this methodology to real-valued functionals on Hilbert space. We define an operator which coincides with the active subspace matrix when applied to a Euclidean space. We show that many of the desirable properties of Active Subspace analysis extend directly to the infinite dimensional setting. We also propose a Monte Carlo procedure and discuss its convergence properties. Finally, we deploy this methodology to create visualizations and improve modeling and optimization on complex test problems.

Active Subspaces in Infinite Dimension

TL;DR

The paper addresses dimension reduction for functionals in infinite dimensions by extending the Active Subspace concept to Hilbert spaces. It defines the active-subspace operator , proves its spectral properties (compact, self-adjoint, trace-class), and derives the leading subspace that captures the dominant gradient energy. A computable Monte Carlo estimator is developed, with proven consistency, eigenvalue/eigenfunction convergence, a rate under fourth-moment assumptions, and a Hilbert-space CLT, enabling reliable uncertainty quantification. Applications to PDE-constrained problems demonstrate how the resulting low-dimensional structure supports visualization, surrogate modeling, and Bayesian optimization in function spaces, with an open-source FEniCS-based implementation illustrating practical impact.

Abstract

Active subspace analysis uses the leading eigenspace of the gradient's second moment to conduct supervised dimension reduction. In this article, we extend this methodology to real-valued functionals on Hilbert space. We define an operator which coincides with the active subspace matrix when applied to a Euclidean space. We show that many of the desirable properties of Active Subspace analysis extend directly to the infinite dimensional setting. We also propose a Monte Carlo procedure and discuss its convergence properties. Finally, we deploy this methodology to create visualizations and improve modeling and optimization on complex test problems.

Paper Structure

This paper contains 17 sections, 9 theorems, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Sensitivity Analysis in Infinite Dimension
  4. The Noncommutativity of Discretization
  5. PDE-Constrained Optimization
  6. Distributed Poisson Control
  7. Laminar Jet
  8. Kirigami Electronics
  9. Active Subspaces in Hilbert Space
  10. Monte Carlo Estimation of $\mathcal{C}$
  11. Applications
  12. Exploratory Analysis and Visualization
  13. Surrogate Modeling
  14. Bayesian Optimization
  15. Discussion
  16. ...and 2 more sections

Key Result

Proposition 1

There exists orthonormal eigenfunctions $\{w_i\}_{i\ge1}\subset\mathcal{H}$ and a nonincreasing sequence $\lambda_1\ge\lambda_2\ge\cdots\ge0$ such that

Figures (4)

  • Figure 1: Random input functions (top row) and corresponding gradient functions (bottom row).
  • Figure 2: Visualization with Active Subspaces. Each row corresponds to a function. Top row: Monte-Carlo estimates of first 10 eigenvalues of active subspace operator. Middle rows: Estimate for first and second eigenfunctions. Bottom row: Projection of samples along first two eigenfunctions (colored points); surface gives a Gaussian process conditional mean estimate of the $L^2$-optimal surrogate fit using hetGPyo2025hetgpy.
  • Figure 3: Active Subspace for functional KNN Regression. Cross Validation MSE comparison for KNN with standard $L^2$ distance versus Euclidean distance along active subspace projection.
  • Figure 4: Functional Optimization in Active Subspace. Compares searching a random vs active subspace; y-axis gives Best Observed Value. Solid line gives median and dotted $10^{th}$ and $90^{th}$ percentile.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Proposition 1: Spectral decomposition of $\mathcal{C}$
  • proof
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2: Hilbert-space MSE bound for the active-subspace surrogate
  • proof
  • Theorem 3: Active directions equalized by $\mathcal{C}^{1/2}$
  • ...and 12 more