# Likelihood-informed dimension reduction for nonlinear inverse problems

arXiv: https://arxiv.org/abs/1403.4680 · HTML: https://mlsanity.com/paper/1403.4680/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2014.

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

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- [Differentiable Inverse Modeling with Physics-Constrained Latent Diffusion for Heterogeneous Subsurface Parameter Fields](https://mlsanity.com/paper/2512.22421/) (2025)
- [The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems](https://mlsanity.com/paper/2606.04804/) (2026)
- [Deep importance sampling using tensor trains with application to a priori and a posteriori rare event estimation](https://mlsanity.com/paper/2209.01941/) (2022)
- [Bi-level Guided Diffusion Models for Zero-Shot Medical Imaging Inverse Problems](https://mlsanity.com/paper/2404.03706/) (2024)
- [Accelerating Hamiltonian Monte Carlo for Bayesian Inference in Neural Networks and Neural Operators](https://mlsanity.com/paper/2507.14652/) (2025)
- [Rotated Mean-Field Variational Inference and Iterative Gaussianization](https://mlsanity.com/paper/2510.07732/) (2025)
- [Multi-Fidelity Delayed Acceptance: hierarchical MCMC sampling for Bayesian inverse problems combining multiple solvers through deep neural networks](https://mlsanity.com/paper/2512.16430/) (2025)
- [Conditional Sampling via Wasserstein Autoencoders and Triangular Transport](https://mlsanity.com/paper/2604.02644/) (2026)
- [PIS: A Generalized Physical Inversion Solver for Arbitrary Sparse Observations via Set Conditioned Flow Matching](https://mlsanity.com/paper/2512.13732/) (2025)
- [Amortized quadrature for posterior expectations in inverse problems](https://mlsanity.com/paper/2606.15871/) (2026)
- [Priors learned from legacy reconstructions inherit undetectable overconfidence](https://mlsanity.com/paper/2607.21721/) (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)
- [SCOPE Before You Sample: Conformal Scores Guide Posterior Exploration for Fast and Calibrated Bayesian Predictive Inference](https://mlsanity.com/paper/2610.17146/) (2026)

## Abstract (arXiv)

The intrinsic dimensionality of an inverse problem is affected by prior information, the accuracy and number of observations, and the smoothing properties of the forward operator. From a Bayesian perspective, changes from the prior to the posterior may, in many problems, be confined to a relatively low-dimensional subspace of the parameter space. We present a dimension reduction approach that defines and identifies such a subspace, called the "likelihood-informed subspace" (LIS), by characterizing the relative influences of the prior and the likelihood over the support of the posterior distribution. This identification enables new and more efficient computational methods for Bayesian inference with nonlinear forward models and Gaussian priors. In particular, we approximate the posterior distribution as the product of a lower-dimensional posterior defined on the LIS and the prior distribution marginalized onto the complementary subspace. Markov chain Monte Carlo sampling can then proceed in lower dimensions, with significant gains in computational efficiency. We also introduce a Rao-Blackwellization strategy that de-randomizes Monte Carlo estimates of posterior expectations for additional variance reduction. We demonstrate the efficiency of our methods using two numerical examples: inference of permeability in a groundwater system governed by an elliptic PDE, and an atmospheric remote sensing problem based on Global Ozone Monitoring System (GOMOS) observations.