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Deep Bayesian Inversion

Jonas Adler, Ozan Öktem

arXiv:1811.05910v1stat.MLcs.LGmath.STstat.TH

TL;DR

Bayesian uncertainty quantification for large-scale inverse problems remains challenging despite its importance for decision making. This paper introduces GAN-based posterior sampling and direct neural estimation, demonstrating promising reconstructions and a greater than 0.95 probability for a suspected liver spot on 2%-dose clinical 3D helical CT.

  • Problem

    Bayesian uncertainty quantification is challenging for ill-posed, high-dimensional inverse problems, even though recovered parameters inform decision making.

    Source: §1 · pp. 1–2

  • Method

    The paper presents Deep Posterior Sampling, which uses a GAN to sample from the posterior, and Deep Direct Estimation, which computes an estimator with a neural network.

    Source: §7 · p. 11

  • Results

    On 2% normal-dose clinical 3D helical CT, both approaches yielded promising posterior means and estimated greater than 0.95 probability that the suspected liver spot exists.

    Source: §5.2 · p. 9, §7 · p. 11

  • Takeaways & Limitations

    The authors conclude these computations can be performed in a timely manner for large-scale clinical 3D helical CT reconstruction.

    Source: §7 · p. 11

    Training generative models for posterior sampling remains complicated and requires substantial fine-tuning and tricks.

    Source: §8 · p. 12

Abstract

from arXiv · show

Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.

1 Introduction

Inverse problems require reliable recovery of hidden parameters from noisy observations, but ill-posedness and large scale make this difficult. Bayesian inversion can support recovery and uncertainty analysis, yet its computational demands limit use in imaging.

1 Introduction

  • Noisy indirect observations make hidden model parameters difficult to recover reliably, especially when inverse problems are ill-posed and large-scale.Ill-posedness means small data errors can cause large parameter errors, and multiple parameter values may fit the observations.Source: §1 · p. 1
  • Bayesian inversion supports parameter estimation and uncertainty quantification, but specifying priors and data measures and exploring the posterior are challenging.The likelihood often comes from a physics-based simulator, while the prior accounts for information about the unknown parameter.Source: §1 · p. 2
  • In 3D CT, handcrafted human-anatomy priors are lacking, and the 512^3-dimensional parameter space makes timely posterior exploration challenging.Current roughness or sparsity priors suppress oscillations at the expense of finer detail; CT data are at least the same order of magnitude as the parameter space.Source: §1 · p. 2

2 Statistical Approach to Inverse Problems

Bayesian inversion characterizes the possible solutions to an inverse problem and their probabilities, providing a framework for uncertainty quantification. Its practical use is limited by difficult prior and data modeling and the computational cost of exploring complex posteriors.

2 Statistical Approach to Inverse Problems

  • The posterior distribution describes possible solutions to an inverse problem and their probabilities, making it essential for uncertainty quantification.Different posterior estimators provide different reconstructions, while Bayesian credible sets offer a framework for uncertainty quantification.Source: §2 · p. 2
  • The posterior has no closed-form expression, and practical Bayesian inversion must address prior selection, data-distribution specification, and computationally feasible posterior exploration.The section identifies these as key challenges to realizing Bayesian inversion’s advantages without access to the full posterior.Source: §2 · pp. 2–3

2.1 Choosing a prior model

  • Handcrafted priors mainly encode regularity such as roughness or sparsity, and their samples poorly capture the a priori information needed to represent natural clinical images.Figure 1 contrasts samples from common imaging priors with typical clinical CT images.Source: §2.1 · p. 3, Figure 1 · p. 3

2.2 Computational feasibility

  • High-dimensional posterior sampling is often too slow or depends on simplifying assumptions, so large-scale inverse problems commonly use a single MAP estimate that omits information in the full posterior.Access to the full posterior could support decisions such as testing whether a tumor is an image artifact.Source: §2.2 · pp. 3–4

3 Contribution

The paper introduces two computationally feasible deep-learning frameworks for uncertainty quantification in inverse problems, using supervised examples to represent the prior and data distribution. It demonstrates them on ultra-low-dose 3D helical CT.

Contribution

  • Both frameworks make uncertainty quantification computationally feasible by representing the prior and data distribution implicitly through supervised examples rather than handcrafted distributions.They are based on generative adversarial networks and are adaptable to inverse problems.Source: §3 · p. 4

Deep Posterior Sampling

  • Deep Posterior Sampling uses a novel conditional Wasserstein GAN discriminator to sample from high-dimensional Bayesian posteriors for statistical analysis.The framework applies in principle to any inverse problem with relevant training data.Source: §3 · p. 4

Deep Direct Estimation

  • Deep Direct Estimation trains a neural network with a chosen loss to compute a pre-determined estimator directly, faster than posterior sampling.The approaches should give similar quantitative results when evaluating the same estimator.Source: §3 · p. 4

Demonstration

  • The authors demonstrate computational feasibility in clinical ultra-low-dose CT by computing estimators and performing a hypothesis test.The imaging application is ultra-low-dose 3D helical CT.Source: §3 · p. 4

4 Deep Bayesian Inversion

Deep Bayesian inversion offers two approaches to posterior analysis: Deep Posterior Sampling learns to generate posterior samples, while Deep Direct Estimation learns estimators directly. Both use supervised training data to make their objectives tractable.

4.1 Deep Posterior Sampling

  • Supervised training pairs let the conditional WGAN objective be estimated by averaging over data, enabling a generator to approximate the posterior for given measurements.The Wasserstein-1 formulation is rewritten as an expectation over the joint law, avoiding direct evaluation of the unknown posterior and data distribution.Source: §4.1 · p. 5
  • A novel conditional mini-batch discriminator addresses mode collapse in the WGAN, while a gradient penalty softly enforces the discriminator’s 1-Lipschitz condition.Without the mini-batch discriminator, the generator may fail to use its noise input.Source: §4.1 · pp. 5–6

4.2 Deep Direct Estimation

  • Deep Direct Estimation trains separate neural networks to approximate the conditional mean and point-wise variance from supervised data, without generating posterior samples.Each estimator can be expressed as a minimizer of an expectation, which is replaced by an average over training data.Source: §4.2 · p. 6
  • Direct estimation is significantly faster than posterior sampling, but less flexible because each estimator requires its own specifically trained neural network.The two approaches should coincide when computing the same estimator.Source: §4.2 · p. 6

5 Numerical Experiments

The experiments evaluate posterior sampling and Direct Estimation for clinical 3D CT imaging. The setup pairs normal-dose images with filtered back-projection reconstructions from ultra-low-dose CT data.

5 Numerical Experiments

  • The study evaluates posterior sampling for drawing from the posterior and Direct Estimation for computing mean and point-wise variances in clinical 3D CT imaging.Source: §5 · p. 6

5.1 Set-up

  • Supervised training pairs normal-dose 3D CT images, treated as ground truth, with filtered back-projection reconstructions from associated ultra-low-dose CT data.Source: §5.1 · pp. 6–7
  • Using filtered back-projection as preprocessing simplifies neural-network architecture choices and poses no limitation in the infinite-data theoretical setting.Source: §5.1 · p. 7

Training data

The study trains and evaluates Bayesian inversion methods on ultra-low-dose CT data derived from the Mayo Clinic challenge, then uses posterior estimates to assess a suspected liver spot. Both approaches estimate a greater than 95% probability that the spot is present, though direct estimation underestimates variance.

Training data

  • Nine of ten Mayo Clinic Low Dose CT challenge scans were used for training and one for evaluation; ultra-low-dose inputs used about 10% of full data with added Poisson noise to represent 2% of normal dose.Filtered back projection (FBP) produced the ultra-low-dose CT images.Source: Training data · p. 7
  • The operators use multi-scale residual neural networks applied slice-wise, with tomography implemented in ODL and ASTRA and learning components in TensorFlow.The slice-wise application was for computational reasons.Source: Training data · p. 7

5.2 Results

  • 40 s per slice for posterior sampling with 1,000 samples versus 80 ms for direct estimation to compute the conditional mean and point-wise standard deviation.Both approaches are described as computationally feasible.Source: §5.2 · p. 8
  • The methods’ conditional means and standard deviations agree well, while direct estimation underestimates variance and posterior sampling gives a more realistic variance estimate.The mean is highly smoothed; uncertainty is high near high-contrast boundaries and white blobs, with background uncertainty around 20 HU from reference-image noise.Source: §5.2 · pp. 8–9, Figure 3 · p. 8
  • p > 0.95 under both posterior sampling and direct estimation for the liver spot’s contrast exceeding 10 HU, supporting its presence at at least 95% significance.The estimate uses 1,000 posterior samples; the direct-estimation distribution assumes independent pixels.Source: §5.2 · p. 9, Figure 4 · p. 9

6 Related Work

Related work spans deep networks that estimate Bayes solutions, generative and MCMC approaches to posterior sampling, and approximations for uncertainty quantification, each with limitations for large-scale inverse problems.

6 Related Work

  • Deep-learning reconstruction uses post-processing [35] or integrated networks [5] [6] [29] [28], often minimizing a loss; prior GAN applications omit generator randomness [67] [46].These loss-based methods amount to direct Bayes estimation, often of the conditional mean under squared L2 loss; direct estimation instead specifies the estimator first. The cited GAN approaches therefore do not fully realize posterior sampling.Source: §6 · pp. 9–10
  • Conditional generative models sample simple posteriors but generally omit physics-driven likelihoods, while MCMC remains computationally unfeasible for large-scale inverse problems such as 3D clinical CT.An approach using variational auto-encoders samples possible segmentations given CT images [41]. MCMC methods are generally rooted in mathematical theory but limited to cases with closed-form priors [11].Source: §6 · p. 10
  • Asymptotic posterior theory assumes uninformative priors; Gaussian-process methods [55] are demanding, neural processes [23] learn broader priors, and MAP credible sets [57, 54] require handcrafted priors.Asymptotic characterizations do not hold for finite data; Gaussian-process models make linear inverse problems tractable when likelihood and prior are Gaussian, but can be costly and difficult to design priors for.Source: §6 · p. 10

7 Conclusions

The paper presents Deep Posterior Sampling and Deep Direct Estimation to enable Bayesian inversion for large-scale inverse problems, demonstrating uncertainty estimation and decision support in ultra-low-dose clinical 3D helical CT.

7 Conclusions

  • Deep Posterior Sampling uses a GAN to sample from the posterior, while Deep Direct Estimation computes an estimator with a neural network, addressing the computational challenge of large-scale Bayesian inversion.Traditional Bayesian inversion requires specifying a prior and, depending on the estimator, the probability of data; exploring the posterior remains computationally challenging.Source: §7 · p. 11
  • At 2% of normal dose, both approaches compute posterior means and point-wise standard deviations for clinical 3D helical CT, while credible sets support testing a suspected liver dark spot.The authors describe the posterior mean reconstruction quality as promising at this dose.Source: §7 · p. 11
  • The authors claim this is the first timely computation for large-scale inverse problems such as clinical 3D helical CT, but clinical use requires trials before image-guided decision making.They identify integrating imaging with clinical decision making while accounting for uncertainty as an advantage.Source: §7 · p. 11
  • Posterior sampling extends beyond specific estimators such as MAP or conditional mean and beyond tomography, given sufficient good supervised training data, and may enable new ways to integrate reconstruction with decision making.The authors frame the approach as applicable to essentially any inverse problem under that training-data condition.Source: §7 · p. 11

8 Discussion and Outlook · A The Wasserstein 1-distance

The discussion identifies open questions about supervised data, posterior convergence, likelihood integration, and training, while the appendix states Wasserstein 1-distance characterizations used in the method.

8 Discussion and Outlook

  • The amount of supervised data needed to resolve the posterior or estimator to a target accuracy is unknown, and standard Bayesian theory assumes an asymptotically non-informative prior.Current theory instead emphasizes an asymptotic regime with indefinitely increasing information in the data.Source: §8 · pp. 11–12
  • Explicit likelihood knowledge may improve results: FBP inputs are theoretically sufficient for posterior samples [4, section 8], while measured-data methods perform better [5] [6].The authors propose conditional WGANs with CNN architectures incorporating handcrafted likelihoods, such as learned iterative reconstruction, especially when supervised data or retraining opportunities are limited.Source: §8 · p. 12
  • Direct estimators trained easily, whereas posterior-sampling generative models required substantial fine-tuning and tricks.The authors hope future generative-model research will make training less complicated.Source: §8 · p. 12

A The Wasserstein 1-distance

  • The Wasserstein 1-distance is presented as a metric on probability measures over a measurable separable Banach space, restricted to finite-distance measures and coupling measures with specified marginals.The coupling family has p and q as its marginals.Source: A The Wasserstein 1-distance · p. 14
  • The Kantorovich–Rubinstein dual characterization expresses the Wasserstein 1-distance using real-valued 1-Lipschitz functions.A function D is 1-Lipschitz when its value differences are bounded by the distance between its inputs.Source: A The Wasserstein 1-distance · p. 14
  • Because enforcing the Lipschitz constraint directly can be difficult, the appendix uses a gradient characterization; in the L2 Hilbert-space setting, the Fréchet derivative is the Hilbert-space gradient.The gradient characterization follows [27] [3].Source: A The Wasserstein 1-distance · p. 14

B Individual Posterior Samples … C.2 A novel discriminator for conditional WGAN

The paper examines posterior samples, derives the conditional WGAN objective, and introduces a conditional mini-batch discriminator to reduce mode collapse when training data provide only one model sample per observation.

B Individual Posterior Samples

  • Generated posterior samples take approximately 40 ms each and look realistic, but some liver blood vessels appear as misplaced white blobs influenced by contrast-enhanced training images.The samples are described as practically indistinguishable from ground truth to an untrained observer.Source: B Individual Posterior Samples · p. 14, Figure 5 · p. 15

C Theory of Deep Bayesian Inversion

C.1 Derivation of conditional WGAN

  • The conditional WGAN derivation uses Wasserstein 1-distance to reformulate the objective without the inaccessible posterior or unknown data distribution, replacing their role with an expectation over the joint law.Supervised training data can provide an empirical counterpart for that joint law; the derivation from the earlier objective involves no approximations.Source: §C.1 · pp. 15–17
  • A generator maps latent noise and an observation to an image, and the empirical training objective can be optimized with stochastic gradient descent to approximate posterior samples.The latent noise is drawn from a simple distribution that can be sampled computationally efficiently.Source: §C.1 · p. 17

C.2 A novel discriminator for conditional WGAN

  • With the original training formulation, the generator tends to ignore latent randomness and repeatedly produce the single training sample available for each observation.This mode collapse is associated with low inter-sample variance and a noisy conditional mean image.Source: §C.2 · pp. 18–19
  • The conditional mini-batch discriminator uses unordered pairs of model and generated samples, making the learned generator less likely to suffer mode collapse.It is designed for Bayesian inversion, where training data provide only one model parameter for each observation.Source: §C.2 · p. 19
  • The generalized training objective has the same solutions as the original objective, under the stated extension of Wasserstein distance to positive Radon measures of equal mass.The proof relies on translation invariance of the Wasserstein metric.Source: §C.2 · p. 19

C.3 Deep Direct Estimation · D Implementation Details

Deep direct estimation trains neural networks to approximate non-randomized estimators, including conditional means and point-wise variances, without drawing posterior samples. The approach is easier to train and faster than posterior sampling, but requires estimator-specific networks and applies only to certain estimators.

C.3 Deep Direct Estimation

  • Direct estimation uses an appropriately trained neural network to approximate a wide range of non-randomized decision rules, unlike posterior sampling, which computes estimators empirically from WGAN samples.The intended applications include uncertainty quantification.Source: §C.3 · p. 20
  • Without the conditional WGAN discriminator, mode collapse produces practically no inter-sample variability and grossly underestimates point-wise standard deviation.The comparison indicates that the conditional discriminator is essential for posterior sampling.Source: Figure 6 · p. 21, Figure 7 · p. 22
  • Squared-error Bayes-risk minimization gives the conditional mean, which neural-network MSE training approximates; 1-norm loss gives the median, while finite-dimensional Bregman losses are also covered.This result also starts the study of maximum a posteriori versus conditional-mean estimates [14].Source: §C.3 · p. 22
  • Point-wise conditional variance can be estimated by rewriting it as a minimizer of an expected scalar objective and training a neural network on supervised data approximating the joint distribution.The resulting estimator is parameterized by a neural network mapping data to variance estimates.Source: §C.3 · pp. 22–23
  • Direct estimation is easier to train than GAN-based posterior sampling and roughly an order of magnitude faster when sufficient posterior samples are needed.Evaluating the trained network takes about as much computation as generating one posterior sample.Source: §C.3 · p. 23
  • Direct estimation requires a separate network for each estimator, limiting patient-specific questions unknown during training, and applies only to estimators expressible as expected scalar-objective minimizers.Approximating posteriors through Edgeworth expansions is possible in principle, but the computations quickly erode the method’s training and computational advantages.Source: §C.3 · pp. 23–24
  • The section’s results and proofs are not fully rigorous in function spaces, although the proofs are straightforward in finite-dimensional spaces.A function-space treatment of Proposition 3 would require higher-moment theory for Banach-space-valued random variables.Source: §C.3 · p. 24

D.1 Training data … E Handcrafted Priors

The paper describes the CT data, neural-network architectures, and training procedures used for Bayesian inversion, and reports that several handcrafted priors approach a Gaussian random-field prior as discretization is refined. It also notes training overfit and that better results may be possible with more data.

D.1 Training data

  • Training data comprised ten normal-dose abdomen CT scans from the Mayo Clinic Low Dose CT challenge, acquired on a Siemens SOMATOM Definition AS+ scanner.Patients predominantly had liver and lung cancer.Source: §D.1 · p. 24
  • Ultra-low-dose inputs were generated by subsampling the CT data and adding Poisson noise to simulate scans at 2% of normal dose.The authors did not model electron noise, despite its significance at these dose levels.Source: §D.1 · p. 24

D.2 Neural networks

  • The generator and direct estimators use a U-Net-like architecture with downsampling, residual blocks, upsampling, and skip connections; the discriminator ends at 8 × 8 pixels with fully connected layers.Source: §D.2 · p. 25

D.3 Training

  • WGAN training alternated one generator step with five discriminator steps, used a gradient penalty, and trained all networks with ADAM for 50,000 steps.The discriminator loss also included drift and weight-decay penalties; the optimization used β1 = 0.5 and β2 = 0.9.Source: §D.3 · pp. 26–27
  • Despite data augmentation and regularization, the networks overfit during training, and the authors expect more data could improve results.Source: §D.3 · p. 27

E Handcrafted Priors

  • Several handcrafted priors, including total variation, converge weakly toward a Gaussian free field as discretization is refined, with little effect on Bayesian inversion solutions compared with a Gaussian random-field prior.The passage attributes differences in MAP solutions largely to a small set of relatively unlikely images.Source: E Handcrafted Priors · p. 27
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