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Latent Space Inference via Paired Autoencoders

Emma Hart, Bas Peters, Julianne Chung, Matthias Chung

TL;DR

The paper addresses inverse problems with observational inconsistencies by introducing PAIR, a paired autoencoder framework that links a parameter-space autoencoder and an observation-space autoencoder via learned latent mappings, enabling optimization directly in latent space. Forward modeling is expressed as $A:\mathcal X\to\mathcal Y$, with $A=Q\circ F$, and latent-space inversion proceeds by solving $\hat{\mathbf z}_x \in \arg\min_{\mathbf z_x} \| (d_y \circ m^{\to})(\mathbf z_x) - \mathbf y \|$ with $\hat{\mathbf x} = d_x(\hat{\mathbf z}_x)$ (or equivalently in $\mathcal Z_{\mathcal Y}$). For incomplete data, the objective uses a projection $P$ so that $\hat{\mathbf z}_x \in \arg\min_{\mathbf z_x} \| (P \circ d_y \circ m^{\to})(\mathbf z_x) - \mathbf y_{\text{sub}} \|$. The paper provides linear-case closed-form solutions and residual/error bounds under Lipschitz and restricted bi-Lipschitz assumptions, and demonstrates the approach on CT tomography and cross-well seismic data, where PAIR+LSI maintains reconstruction quality when data are missing or corrupted and offers practical uncertainty diagnostics.

Abstract

This work describes a novel data-driven latent space inference framework built on paired autoencoders to handle observational inconsistencies when solving inverse problems. Our approach uses two autoencoders, one for the parameter space and one for the observation space, connected by learned mappings between the autoencoders' latent spaces. These mappings enable a surrogate for regularized inversion and optimization in low-dimensional, informative latent spaces. Our flexible framework can work with partial, noisy, or out-of-distribution data, all while maintaining consistency with the underlying physical models. The paired autoencoders enable reconstruction of corrupted data, and then use the reconstructed data for parameter estimation, which produces more accurate reconstructions compared to paired autoencoders alone and end-to-end encoder-decoders of the same architecture, especially in scenarios with data inconsistencies. We demonstrate our approaches on two imaging examples in medical tomography and geophysical seismic-waveform inversion, but the described approaches are broadly applicable to a variety of inverse problems in scientific and engineering applications.

Latent Space Inference via Paired Autoencoders

TL;DR

The paper addresses inverse problems with observational inconsistencies by introducing PAIR, a paired autoencoder framework that links a parameter-space autoencoder and an observation-space autoencoder via learned latent mappings, enabling optimization directly in latent space. Forward modeling is expressed as , with , and latent-space inversion proceeds by solving with (or equivalently in ). For incomplete data, the objective uses a projection so that . The paper provides linear-case closed-form solutions and residual/error bounds under Lipschitz and restricted bi-Lipschitz assumptions, and demonstrates the approach on CT tomography and cross-well seismic data, where PAIR+LSI maintains reconstruction quality when data are missing or corrupted and offers practical uncertainty diagnostics.

Abstract

This work describes a novel data-driven latent space inference framework built on paired autoencoders to handle observational inconsistencies when solving inverse problems. Our approach uses two autoencoders, one for the parameter space and one for the observation space, connected by learned mappings between the autoencoders' latent spaces. These mappings enable a surrogate for regularized inversion and optimization in low-dimensional, informative latent spaces. Our flexible framework can work with partial, noisy, or out-of-distribution data, all while maintaining consistency with the underlying physical models. The paired autoencoders enable reconstruction of corrupted data, and then use the reconstructed data for parameter estimation, which produces more accurate reconstructions compared to paired autoencoders alone and end-to-end encoder-decoders of the same architecture, especially in scenarios with data inconsistencies. We demonstrate our approaches on two imaging examples in medical tomography and geophysical seismic-waveform inversion, but the described approaches are broadly applicable to a variety of inverse problems in scientific and engineering applications.
Paper Structure (9 sections, 1 theorem, 36 equations, 7 figures, 1 table)

This paper contains 9 sections, 1 theorem, 36 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction and Background
  2. Background
  3. Latent Space Inference using PAIR
  4. Latent Space Inference with Linear Constructions
  5. Residual and Error Bounds for Latent Space Inference
  6. Numerical Results and Comparisons
  7. Example 1: CT Imaging
  8. Example 2: Cross-Well Seismic Inversion
  9. Conclusions

Key Result

Proposition 1

Let all the assumptions of Statement fact:proj_res hold. Let $\mathcal{S}\subseteq \mathcal{Y}$ be a subset such that $\mathbf{y}\in\mathcal{S}$ and $d_y(\widehat{\mathbf{z}})\in\mathcal{S}$, and assume that $P:\mathcal{Y}\to\mathcal{Y}_{\rm sub}$ satisfies a restricted bi-Lipschitz condition on $\m such that for all $\mathbf{w}_1, \mathbf{w}_2 \in \mathcal{S}$. Additionally, let $d_x$, $m^\gets$

Figures (7)

  • Figure 1: Example reconstructions of PAIR, end-to-end encoder decoders, and PAIR + LSI from sinogram $\mathbf{Y}$, with different projections. Each column header shows the input sinogram that has been corrupted with random or blocks of missing angles. The target $\mathbf{X}$ is representative of the samples on which all networks were trained.
  • Figure 2: Reconstructions of an out-of-distribution test sample using PAIR, end-to-end encoder decoders, and PAIR + LSI from different observations of $\mathbf{Y}$. Each column header shows the input sinogram that has been corrupted with random or blocks of missing angles, and the OOD target $\mathbf{X}$ is provided for comparison.
  • Figure 3: PAIR out-of-distribution metrics for in-distribution samples with full observational data, samples with a block of masked angles, and samples with a block of masked angles reconstructed using PAIR+LSI.
  • Figure 4: Seismic FWI example: (a) sample velocity models, (b) full (uncorrupted) data for three sources/channels.
  • Figure 5: Top row: Observed data for one source, with full data (left) and with $90\%$ random missing receivers (right). Middle row: Five samples of data reconstructions using PAIR+LSI. Bottom row: Average and standard deviation of data reconstructions. Data reconstructions in the physical space appear as an intermediate quantity only.
  • ...and 2 more figures

Theorems & Definitions (4)

  • proof
  • Proposition 1
  • proof
  • proof