Bayesian imaging inverse problem with scattering transform

Technical Summary: Bayesian Imaging Inverse Problem with Scattering Transform

Problem Formulation
The paper addresses the challenge of solving Bayesian imaging inverse problems in astrophysics and cosmology, specifically within "low-data regimes" where only a single contaminated observation ($d_0$) is available. The core difficulty lies in recovering an underlying physical field ($s_0$) from data produced by a known but often stochastic or non-invertible forward operator ($F$), such that $d_0 = F(s_0)$.

Traditional Bayesian approaches require a prior distribution $p(s)$ encoding physical knowledge about the signal. However, for complex, non-Gaussian astrophysical fields (e.g., large-scale structure, Galactic dust), specifying a physically motivated prior is difficult, and data-driven priors fail when training data is scarce. The authors frame the problem as characterizing the posterior distribution $p(s | d_0)$ without relying on external priors or large training datasets, focusing instead on non-Gaussian, regular fields.

Methodology
The proposed approach shifts the inference objective from the high-dimensional pixel space ($n \approx 10^5$) to a low-dimensional manifold of Scattering Transform (ST) statistics ($l \approx 10^2$).

1. Scattering Transform Representation: The method utilizes ST statistics, $\phi(s)$, which characterize non-Gaussian processes by capturing interactions between oriented scales. These statistics are used to construct efficient maximum entropy generative models, $p_{\mu_S}(s)$, parametrized by the ST statistics $\mu_S$.
2. Dimensionality Reduction: Instead of inferring the posterior over fields $p(s | d_0)$, the authors infer the posterior over the ST statistics parameters: $p(\mu_S | \phi(d_0))$. This reduces the problem to finding a generative model consistent with the observed data's statistics.
3. Likelihood Approximation: The forward operator in ST space, mapping parameters $\mu_S$ to data statistics $\phi(d)$, is analytically intractable due to the sampling required from generative models. The authors approximate this likelihood using a Gaussian linear model derived via Taylor expansion: $p(\phi(d) | \mu_S) \sim \mathcal{N}(A\mu_S + b | \Sigma)$.
4. Iterative Adaptive Algorithm: To estimate the parameters ($A, b, \Sigma$) of this Gaussian likelihood and the resulting posterior, the authors propose a sequential estimation algorithm:
   • Initialization: A narrow Gaussian proposal $q_0(\mu_S)$ is centered on the ST statistics of a reference map $\tilde{s}$, obtained by optimizing white noise to minimize the distance between its transformed statistics and the observed data statistics.
   • Sequential Refinement: The proposal distribution is iteratively updated by combining the previous proposal with the estimated likelihood. The covariance is scaled by a factor of 2 at each step to prevent artificial overconfidence from conditioning on the same observation multiple times.
   • Convergence: The process stops when the proposal distribution converges to a fixed point, coinciding with the posterior distribution under a uniform prior and linear Gaussian likelihood.

Key Contributions

• Novel Inference Framework: The paper introduces a Bayesian framework that performs inference in the space of Scattering Transform statistics rather than pixel space, enabling tractable solutions for non-Gaussian fields without external priors.
• Adaptive Algorithm: It proposes a specific iterative algorithm that approximates the posterior distribution of generative model parameters using a linear Gaussian likelihood approximation, avoiding the computational cost of training neural density estimators in high-dimensional spaces.
• Single-Image Reconstruction: The method is designed to operate in a regime with only a single observation, bypassing the need for large training datasets typically required by deep learning approaches.

Results
The method was validated using a column density field from the Quijote cosmological simulations (large-scale structure) and a mock instrumental forward operator consisting of white Gaussian noise and random elliptical masks.

• Statistical Inference: The posterior samples of the ST statistics accurately reproduced the statistical properties (power spectrum and one-point probability distribution function) of the true underlying field $s_0$.
• Deterministic Reconstruction: By sampling from the inferred generative models, the authors generated maps that were visually indistinguishable from the observation when passed through the forward operator. The generated fields themselves were consistent with Large Scale Structure (LSS) maps, exhibiting variability primarily at the smallest, noise-dominated scales.
• No External Priors: The reconstruction was achieved without any external prior distribution for the field of interest, relying solely on the ST-based generative model structure.

Significance and Claims
The paper claims that Scattering Transform statistics provide an effective representation for solving complex imaging inverse problems in challenging low-data regimes. The approach demonstrates that accurate statistical inference and deterministic signal reconstruction are possible for non-Gaussian astrophysical signals even when:

1. Only a single contaminated image is available.
2. No physically motivated prior or realistic training dataset exists.

The authors state that this work opens the way to new applications for non-Gaussian astrophysical and cosmological signals where little or no prior modeling is currently available, offering a generalizable solution beyond the specific Large Scale Structure example tested.