# Certified dimension reduction in nonlinear Bayesian inverse problems

arXiv: https://arxiv.org/abs/1807.03712 · HTML: https://mlsanity.com/paper/1807.03712/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2018.

## Built on by (most cited first, 27 of 27)

- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Projected Wasserstein gradient descent for high-dimensional Bayesian inference](https://mlsanity.com/paper/2102.06350/) (2021)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Derivative-enhanced Deep Operator Network](https://mlsanity.com/paper/2402.19242/) (2024)
- [Learning non-Gaussian graphical models via Hessian scores and triangular transport](https://mlsanity.com/paper/2101.03093/) (2021)
- [Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization](https://mlsanity.com/paper/2205.13098/) (2022)
- [A low-rank ensemble Kalman filter for elliptic observations](https://mlsanity.com/paper/2203.05120/) (2022)
- [Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance](https://mlsanity.com/paper/2207.14387/) (2022)
- [Expected information gain estimation via density approximations: Sample allocation and dimension reduction](https://mlsanity.com/paper/2411.08390/) (2024)
- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Active Slices for Sliced Stein Discrepancy](https://mlsanity.com/paper/2102.03159/) (2021)
- [Stein's method for marginals on large graphical models](https://mlsanity.com/paper/2410.11771/) (2024)
- [Coupled Input-Output Dimension Reduction: Application to Goal-oriented Bayesian Experimental Design and Global Sensitivity Analysis](https://mlsanity.com/paper/2406.13425/) (2024)
- [Dimension reduction via score ratio matching](https://mlsanity.com/paper/2410.19990/) (2024)
- [Transport Quasi-Monte Carlo](https://mlsanity.com/paper/2412.16416/) (2024)
- [Deep importance sampling using tensor trains with application to a priori and a posteriori rare event estimation](https://mlsanity.com/paper/2209.01941/) (2022)
- [Comparison of parallel SMC and MCMC for Bayesian deep learning](https://mlsanity.com/paper/2402.06173/) (2024)
- [Surrogate to Poincar'e inequalities on manifolds for dimension reduction in nonlinear feature spaces](https://mlsanity.com/paper/2505.01807/) (2025)
- [Rotated Mean-Field Variational Inference and Iterative Gaussianization](https://mlsanity.com/paper/2510.07732/) (2025)
- [Copula Active Subspaces I: A Score-Covariance Method for Reduced-Order Non-Gaussian Density Estimation](https://mlsanity.com/paper/2609.36142/) (2026)
- [A local approach to parameter space reduction for regression and classification tasks](https://mlsanity.com/paper/2107.10867/) (2021)
- [Amortized quadrature for posterior expectations in inverse problems](https://mlsanity.com/paper/2606.15871/) (2026)
- [Gradient-free Riemannian Langevin Sampler](https://mlsanity.com/paper/2607.07519/) (2026)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)
- [High-Dimensional Simulation-Based Inference in Latent Spaces](https://mlsanity.com/paper/2609.37381/) (2026)
- [Search Dimension in Unlabeled Projection Pursuit: A Scaling Law for Subspace Restriction](https://mlsanity.com/paper/2609.37917/) (2026)
- [Copula Active Subspaces II: Error Decomposition, A Posteriori Estimation, and Sharpness of the Bounds](https://mlsanity.com/paper/2610.14498/) (2026)

## Abstract (arXiv)

We propose a dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation noise. The likelihood function is approximated by a ridge function, i.e., a map which depends non-trivially only on a few linear combinations of the parameters. We build this ridge approximation by minimizing an upper bound on the Kullback--Leibler divergence between the posterior distribution and its approximation. This bound, obtained via logarithmic Sobolev inequalities, allows one to certify the error of the posterior approximation. Computing the bound requires computing the second moment matrix of the gradient of the log-likelihood function. In practice, a sample-based approximation of the upper bound is then required. We provide an analysis that enables control of the posterior approximation error due to this sampling. Numerical and theoretical comparisons with existing methods illustrate the benefits of the proposed methodology.