Coupled Input-Output Dimension Reduction: Application to Goal-oriented Bayesian Experimental Design and Global Sensitivity Analysis

TL;DR

This work develops a coupled input-output dimension reduction framework for a nonlinear map $G:\mathbb{R}^d\to\mathbb{R}^m$ that jointly reduces input and output subspaces via $X_r=U_r^{\top}X$ and $Y_s=V_s^{\top}Y$. It derives gradient-based upper and lower bounds on the $L^2$-approximation error using Poincaré and Cramér-Rao-like inequalities, expressed through two diagnostic matrices $H_X(V_s)$ and $H_Y(U_r)$, and solves for the optimal subspaces with an alternating eigendecomposition. The framework naturally supports goal-oriented tasks, enabling goal-oriented Bayesian optimal experimental design and goal-oriented global sensitivity analysis by identifying the most informative input or output components from the dominant eigen-directions or diagonal entries of the diagnostic matrices. Numerical experiments on conditioned diffusion and Burgers’ equation illustrate that the derivative-based bounds closely track true information measures and Sobol’ indices, while the coupled reduction outperforms traditional PCA-based approaches in highlighting temporally or spatially relevant modes. The approach provides computable, gradient-based subspace selection that bypasses expensive objective evaluations and combinatorial search, with potential extensions to operator learning and nonlinear goals.

Abstract

We introduce a new method to jointly reduce the dimension of the input and output space of a function between high-dimensional spaces. Choosing a reduced input subspace influences which output subspace is relevant and vice versa. Conventional methods focus on reducing either the input or output space, even though both are often reduced simultaneously in practice. Our coupled approach naturally supports goal-oriented dimension reduction, where either an input or output quantity of interest is prescribed. We consider, in particular, goal-oriented sensor placement and goal-oriented sensitivity analysis, which can be viewed as dimension reduction where the most important output or, respectively, input components are chosen. Both applications present difficult combinatorial optimization problems with expensive objectives such as the expected information gain and Sobol' indices. By optimizing gradient-based bounds, we can determine the most informative sensors and most influential parameters as the largest diagonal entries of some diagnostic matrices, thus bypassing the combinatorial optimization and objective evaluation.

Figures (8)

• Figure 1: Illustration of the inputs and outputs of interest (i.e., goals) within the specified orange time intervals, and their corresponding goal-oriented solutions for the vectors defining the reduced output and input spaces, respectively. Notice that the solutions correctly reflect the time causality of the problem.
• Figure 2: Goal-oriented total Sobol' indices for each index set as compared with the derived derivative-based bounds. The bounds tightly track the true value for the Sobol' indices. The bounds also take into account the time causality of the problem.
• Figure 3: Comparison of subspaces from coupled and PCA-based dimension reduction.
• Figure 4: Convergence of of alternating eigendecomposition for $10$ random initializations.
• Figure 5: Comparison of model error and upper bound convergence across different dimension reduction methods as subspace dimensions $r,\; s$ increase.

Theorems & Definitions (20)

• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• Definition 3.4: Poincaré and Cramér-Rao inequality
• Theorem 3.5
• Proposition 3.6
• Proof 1
• Remark 3.7: Preconditioning
• Remark 3.8: Equality and Closed-form Solution for Affine Models
• Lemma 3.9: Variational characterization of eigenvalues of Hermitian matrices