Structured Column Subset Selection for Bayesian Optimal Experimental Design

TL;DR

This paper extends optimal experimental design for Bayesian inverse problems to multi-way structured variables by mapping the problem to tensor-based column subset selection, introducing algorithms and demonstrating effectiveness across various applications.

Contribution

It introduces a tensor-based extension of column subset selection for structured Bayesian experimental design, with new algorithms and analysis for multi-way design variables.

Findings

Algorithms are effective and scalable for structured design.

Numerical experiments demonstrate success across multiple applications.

Proposed methods outperform traditional approaches in complex scenarios.

Abstract

We consider optimal experimental design (OED) for Bayesian inverse problems, where the experimental design variables have a certain multiway structure. Given $d$ different experimental variables with $m_i$ choices per design variable $1 \le i\le d$, the goal is to select $k_i \le m_i$ experiments per design variable. Previous work has related OED to the column subset selection problem by mapping the design variables to the columns of a matrix $\mathbf{A}$. However, this approach is applicable only to the case $d=1$ in which the columns can be selected independently. We develop an extension to the case where the design variables have a multi-way structure. Our approach is to map the matrix $\mathbf{A}$ to a tensor and perform column subset selection on mode unfoldings of the tensor. We develop an algorithmic framework with three different algorithmic templates, and randomized variants of these algorithms. We analyze the computational cost of all the proposed algorithms and also develop greedy versions to facilitate comparisons. Numerical experiments on four different applications -- time-dependent inverse problems, seismic tomography, X-ray tomography, and flow reconstruction -- demonstrate the effectiveness and scalability of our methods for structured experimental design in Bayesian inverse problems.