Numerical methods in inverse problems

Latest papers

• PosteriorBench: From Point Estimates to Posterior Matching in Evaluating Generative Inverse Solvers

  Jiachen Yao, Zi-Siang Hsu, Xi Deng, Aditi Gupta, Xin Ju, Sally M Benson, Gege Wen, Anima Anandkumar · 18 September 2026

  Generative models are increasingly used to solve scientific inverse problems, but existing evaluations still focus primarily on whether a method can produce a single plausible reconstruction. This is insufficient for ill-posed problems, where multiple solutions may be consistent with the same sparse…

• Fast and Faithful: Principled Conditional Flow Matching for Inverse Problems

  Shirin Shoushtari, Edward P. Chandler, Xiao Shi, Ulugbek S. Kamilov · 14 September 2026

  Flow matching approaches to imaging inverse problems commonly incorporate measurements in two ways. Conditioning-based approaches supply measurement-derived information as a network input, often through concatenation, while inference-guided approaches combine an unconditional velocity field with a s…

• Why Learning Rediscovers the Closed-Form Diagonal Regularizer

  Jeahn Han, Pyojin Kim · 10 September 2026

  We identify a diagonal saturation principle in modal inverse problems: when truncation noise is isotropic, the Bayes-optimal Tikhonov shape is a closed-form power law Gamma_k proportional to lambda_k^|s| set by the prior alone, independent of the domain. Berry's random-wave conjecture decorrelates t…

• Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem

  Yang Zhao, Junxiong Jia, Tao Zhou · 4 September 2026

  This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional mod…

• Spectral Convergence of Random Feature Method in Multiple Dimensions

  Pingbing Ming, Hao Yu · 4 September 2026

  We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operato…

• Matched Queries for Curvature and Density at Branching Junctions

  Ziqi Zhao, Qingjian Ni · 2 September 2026

  At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, …

• Generalized Splines and Gaussian Processes

  Michael Unser · 31 August 2026

  For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting…

• On the Computational and Statistical Efficiency of the Empirical Maximum Entropy on the Mean Method

  Matthew King-Roskamp, Gabriel Rioux, Rustum Choksi, Tim Hoheisel · 31 August 2026

  The Maximum Entropy on the Mean (MEM) method provides a flexible computational framework for solving inverse problems by combining data fidelity with entropy-based regularization. In practice, however, the prior distribution is typically unknown but can be estimated from data, giving rise to the emp…

• Physics-informed learning for the inverse problem in resonant ultrasound spectroscopy

  Alejandro Cubillos Mu\~noz, Manuela Rivas, Julian Rincon · 31 August 2026

  Inferring elastic constants from resonant ultrasound spectra is a nonlinear and typically overdetermined inverse problem based on finite spectral data. We formulate the Rayleigh-Ritz inverse problem as a constrained inverse-isospectral problem on the set of physically admissible elasticity tensors. …

• Active Diffusion-Based Inference for Ill-Posed Inverse Problems under Incomplete Priors

  Jitao Xu, Nobuo Sato, Yaohang Li · 28 August 2026

  Many scientific and engineering applications require estimating unknown parameters from experimentally observable data -- an inverse problem that is inherently challenging due to nonlinearity, noise, and ill-posedness. In this paper, we propose an active diffusion-based inverse problem solver. A DM …

• Learning spatially varying regularisation parameters of low regularity for image reconstruction

  Kostas Papafitsoros, Luca Calatroni, Andreas Kofler · 27 August 2026

  In this chapter, we review and discuss the regularity properties of spatially adaptive regularisation weight functions used in variational image reconstruction. Incorporating such weights into classical model-based regularisers, such as Total Variation (TV) and Total Generalised Variation (TGV), all…

• Sequential operator learning under dependent data

  Rafael Oliveira · 26 August 2026

  Learning operators from sequentially collected data arises in adaptive experimental design, Bayesian optimization, and dynamical-system modelling, where observations may be dependent, and future inputs or sensing operators may depend on preceding data. We derive time-uniform self-normalized concentr…

• Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime

  Hugo Latourelle-Vigeant, Sinho Chewi, Aram-Alexandre Pooladian, John Sous, Theodor Misiakiewicz · 26 August 2026

  Modern score-based generative models have achieved remarkable empirical success in high-dimensional tasks such as image, audio, and video synthesis. These models reduce distribution learning to a sequence of regression problems that, if solved exactly on finite data, would ultimately reproduce the t…

• Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples

  Xinliang Liu, Tong Mao, Jinchao Xu · 10 August 2026

  We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_\beta u=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_\beta$ is a positive elliptic spectral multiplier of order $\beta$. Given a parameter set …

• Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators

  Maximiliano Hertel, Ilja Klebanov, Manuel Schaller, Karl Worthmann · 7 August 2026

  Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO ma…

• Discretization and Statistical Consistency of Functional Flow Matching

  Lennon J. Shikhman · 6 August 2026

  Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove…

• A convergence result of a continuous model of deep learning via a \L{}ojasiewicz--Simon inequality

  Noboru Isobe · 23 July 2026

  We study an idealized training process for deep neural networks in a continuous-depth, mean-field model in which each layer is parameterized by a probability measure on a Euclidean parameter space. The training dynamics are formulated as a Wasserstein-type gradient flow of an objective with a fixed …

• Provable diffusion-based posterior sampling for linear inverse problems via DDIM

  Yuchen Jiao, Na Li, Changxiao Cai, Yuxin Chen, Gen Li · 22 July 2026

  Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving l…

• Statistical inverse learning problems with random observations

  Abhishake Rastogi, Tapio Helin, Nicole M\"ucke · 10 July 2026

  We provide an overview of recent progress in statistical inverse problems with random experimental design, covering both linear and nonlinear inverse problems. Different regularization schemes have been studied to produce robust and stable solutions. We discuss recent results in spectral regularizat…

• Statistical inverse learning and $\ell^1$-regularization

  Abhishake Rastogi, Tatiana A. Bubba, Tapio Helin, Luca Ratti · 9 July 2026

  We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning. The unknown is modeled as an element of $\ell^1$, and observations are generated through a possibly nonlinear forward operator $A:\ell^1\to H$, where $H$ is a vec…

• What Do Flow-Based Inverse Solvers Approximate? A Posterior-Transport View

  Jian Xu, Delu Zeng, John Paisley, Qibin Zhao · 24 June 2026

  A growing family of training-free solvers -- FlowDPS, FLOWER, PnP-Flow and their diffusion ancestors (DPS, DAPS) -- repurpose a pretrained flow-matching prior to solve imaging inverse problems by adding a measurement-guidance term to the deterministic probability-flow ODE. Despite strong empirical r…

• Which Spaces can be Embedded in $L_p$-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy

  Yiping Lu, Daozhe Lin, Qiang Du · 24 June 2026

  In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces. Metric entropy characterizes the information complexity of function spaces and has implications for their approximability and learn…

• Risk Comparisons in Linear Regression: Implicit Regularization Dominates Explicit Regularization

  Jingfeng Wu, Peter L. Bartlett, Sham M. Kakade, Jason D. Lee, Bin Yu · 10 June 2026

  Existing theory suggests that for linear regression problems categorized by capacity and source conditions, gradient descent (GD) is always minimax optimal, while both ridge regression and online stochastic gradient descent (SGD) are polynomially suboptimal for certain categories of such problems. M…

• No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty

  Ronald Katende · 8 June 2026

  Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is a…

• Target localization, identification and sensing using latent symmetries

  David Dukov, Malte R\"ontgen, Bryn Davies · 2 June 2026

  We show that an array of scatterers which has been designed to have latent ("hidden") symmetries can be used as a sensor. We use the capacitance matrix as a canonical model for three-dimensional hybridisation and study how the introduction of an "intruder'' scatterer breaks the latent symmetries. By…