Designing algorithms for solving high-dimensional Bayesian inverse problems directly in infinite‑dimensional function spaces – where such problems are naturally formulated – is crucial to ensure stability and convergence as the discretization of the underlying problem is refined. In this paper, we contribute to this line of work by analyzing a widely used sampler for linear inverse problems: Langevin dynamics driven by score‑based generative models (SGMs) acting as priors, formulated directly in function space. Building on the theoretical framework for SGMs in Hilbert spaces, we give a rigorous definition of this sampler in the infinite-dimensional setting and derive, for the first time, error estimates that explicitly depend on the approximation error of the score. As a consequence, we obtain sufficient conditions for global convergence in Kullback–Leibler divergence on the underlying function space. Preventing numerical instabilities requires preconditioning of the Langevin algorithm and we prove the existence and form of an optimal preconditioner. The preconditioner depends on both the score error and the forward operator and guarantees a uniform convergence rate across all posterior modes. Our analysis applies to both Gaussian and a general class of non‑Gaussian priors. Finally, we present examples that illustrate and validate our theoretical findings.

Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions—the conditional denoising estimator—can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference.

In this paper we introduce and analyze a passive synthetic aperture radar system motivated by space surveillance radar networks for detecting, tracking, imaging, and identifying small debris (aka target) in low-earth orbit. We propose a system with a powerful transmitter on the ground and one or several flying receiver platforms. Each platform can separate the direct signals from the source, coming from below, and the reflected signals, coming from above. In the case of a single receiver platform, the image is formed by cross correlating the Doppler-compensated direct and reflected signals. We show that its resolution is described by the ideal Rayleigh resolution formulas. That is, range resolution is proportional to the transmitter pulse width, and the cross-range resolutions are determined by the synthetic angular cones determined by the target and receiver trajectories. However, the one-receiver imaging modality is not capable of determining both the target location and velocity. In the case of multiple receiver platforms, the image is computed by cross correlating the Doppler-compensated reflected signals. In this case, the image focusing relies on differences of travel times between the moving target and the receivers, and we show that two well separated pairs of receiver platforms are sufficient for determining the target location and velocity. The resolution formulas for this imaging modality are new and different from the Rayleigh resolution limits. They are derived analytically from first principles and are validated with detailed numerical simulations.

International audience

It is known that cross correlation of waves generated by noise sources, propagating in an unknown medium and recorded by a sensor array, can provide information about the medium. In this paper the medium is a three-dimensional small-scale randomly layered medium with slow macroscopic variations. The main objective here is to set forth a framework for analysis of cross correlations of waves generated by noise sources and propagating in such a medium and, moreover, to use this framework to design estimators for macroscale medium features. The noise sources are located at the bottom of a random medium slab and generate a random wave field that is scattered by the rapid random fluctuations of the medium and then recorded at the surface. Taking into account the pressure release boundary conditions at the surface, this situation corresponds to the so-called daylight configuration. The analysis is carried out in the asymptotic framework where the typical wavelength is small compared to the scale of the macroscopic variations of the background medium and large compared to the decoherence length of the microscopic random fluctuations of the medium. It is shown that the cross correlation of the waves recorded at the surface contains statistically stable information about the macroscopic background medium.

This paper presents a study of stability, resolution, and detection for broad-band synthetic aperture (SA) imaging in Rician fading media. The stability condition $BNK^2/(K+1)\gg M$ is derived, where K is the Rician factor, B is the effective number of coherence bands, N is the effective number of array elements, and M is the number of (widely separated) targets. The imaging method is tested numerically with randomly distributed discrete scatterers, and comparisons with the imaging with the full response matrix (RM) are made. The resolution study reveals several interesting effects: First, given the same measurement resources, SA imaging has better resolution performance, although less stable, than RM imaging; second, for both imaging methods, the cross-range resolution measure (i.e., “full width at half maximum”) decreases with the aperture (N fixed) and the probe spacing (the total aperture fixed) while the range resolution increases with both parameters. A statistical scheme is introduced to reduce the uncertainty when the stability regime is not realized.

We develop an expectation-maximization algorithm with local adaptivity for image segmentation and classification. The key idea of our approach is to combine global statistics extracted from the Gaussian mixture model or other proper statistical models with local statistics and geometrical information, such as local probability distribution, orientation, and anisotropy. The combined information is used to design an adaptive local classification strategy that improves the robustness of the algorithm and also keeps fine features in the image. The proposed methodology is flexible and can be easily generalized to deal with other inferred information/quantities and statistical methods/models.