Numerical methods in inverse problems

Neueste Paper

• 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. Juli 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. Juli 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. Juli 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. Juli 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…

• 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. Juni 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. Juni 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. Juni 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. Juni 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…

• A Unifying View of Anchoring via Operator-Side Tikhonov Regularization

  Zihao Chen · 1. Juni 2026

  Anchored fixed point and monotone equation methods, including Halpern iteration, extra anchored gradient, and their relatives, add a vanishing pull toward a reference point to obtain last-iterate guarantees. Existing anchored variants often achieve sharp last-iterate guarantees, but from the update-…

• Measure flow path recovery in Bayes Hilbert spaces

  S. David Mis, Maarten V. de Hoop · 29. Mai 2026

  We study the ill-posed problem of recovering a probability measure flow from finitely many moving localized sensors using a Bayes Hilbert framework. Relative to a fixed reference probability measure, a probability law is represented by its centered log-ratio coordinates, so that an evolving law beco…

• UOTIP: Unbalanced Optimal Transport Map for Unpaired Inverse Problems

  Donggyu Lee, Taekyung Lee, Jaewoong Choi · 21. Mai 2026

  We investigate unpaired image inverse problems, a challenging setting where only independent, non-paired sets of noisy measurements and clean target signals are available for training. We propose a novel inverse problem solver based on Unbalanced Optimal Transport, called Unbalanced Optimal Transpor…

• Randomized Atomic Feature Models for Physics-Informed Identification of Dynamic Systems

  Rajiv Singh, Mario Sznaier, Lennart Ljung · 15. Mai 2026

  We present a physics-informed framework for system identification based on randomized stable atomic features. Impulse responses are represented as random superpositions of stable atoms, namely damped complex exponentials associated with poles sampled inside a prescribed disk. Identification is then …

• Learning Curves and Benign Overfitting of Spectral Algorithms in Large Dimensions

  Weihao Lu, Qian Lin, Yingcun Xia, Dongming Huang · 28. April 2026

  Existing large-dimensional theory for spectral algorithms resolves either the optimally tuned point or the interpolation limit, but leaves the under-regularized regime unexplored. We study the learning curve and benign overfitting of spectral algorithms in the large-dimensional setting where the sam…

• Interpretable Operator Learning for Inverse Problems via Adaptive Spectral Filtering: Convergence and Discretization Invariance

  Hang-Cheng Dong, Pengcheng Cheng, Shuhuan Li · 24. März 2026

  Solving ill-posed inverse problems necessitates effective regularization strategies to stabilize the inversion process against measurement noise. While classical methods like Tikhonov regularization require heuristic parameter tuning, and standard deep learning approaches often lack interpretability…

• Forward and inverse problems for measure flows in Bayes Hilbert spaces

  S. David Mis, Maarten V. de Hoop · 24. März 2026

  We study forward and inverse problems for time-dependent probability measures in Bayes--Hilbert spaces. On the forward side, we show that each sufficiently regular Bayes--Hilbert path admits a canonical dynamical realization: a weighted Neumann problem transforms the log-density variation into the u…

• Laplace-Beltrami Operator for Gaussian Splatting

  Hongyu Zhou, Zorah L\"ahner · 18. März 2026

  With the rising popularity of 3D Gaussian splatting and the expanse of applications from rendering to 3D reconstruction, there comes also a need for geometry processing applications directly on this new representation. While considering the centers of Gaussians as a point cloud or meshing them is an…

• Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss

  Lucas Amoudruz, Sergey Litvinov, Costas Papadimitriou, Petros Koumoutsakos · 6. März 2026

  Inverse problems are crucial for many applications in science, engineering and medicine that involve data assimilation, design, and imaging. Their solution infers the parameters or latent states of a complex system from noisy data and partially observable processes. When measurements are an incomple…

• Infinite dimensional generative sensing

  Paolo Angella, Vito Paolo Pastore, Matteo Santacesaria · 4. März 2026

  Deep generative models have become a standard for modeling priors for inverse problems, going beyond classical sparsity-based methods. However, existing theoretical guarantees are mostly confined to finite-dimensional vector spaces, creating a gap when the physical signals are modeled as functions i…

• Learning sparsity-promoting regularizers for linear inverse problems

  Giovanni S. Alberti, Ernesto De Vito, Tapio Helin, Matti Lassas, Luca Ratti, Matteo Santacesaria · 3. März 2026

  This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution.…

• EquiReg: Equivariance Regularized Diffusion for Inverse Problems

  Bahareh Tolooshams, Aditi Chandrashekar, Rayhan Zirvi, Abbas Mammadov, Jiachen Yao, Chuwei Wang, Anima Anandkumar · 3. März 2026

  Diffusion models represent the state-of-the-art for solving inverse problems such as image restoration tasks. Diffusion-based inverse solvers incorporate a likelihood term to guide prior sampling, generating data consistent with the posterior distribution. However, due to the intractability of the l…

• A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron

  Dan Garber · 3. März 2026

  We consider the problem of minimizing a smooth and convex function over the $n$-dimensional spectrahedron -- the set of real symmetric $n\times n$ positive semidefinite matrices with unit trace, which underlies numerous applications in statistics, machine learning and additional domains. Standard fi…

• Sample-efficient evidence estimation of score based priors for model selection

  Frederic Wang, Katherine L. Bouman · 25. Februar 2026

  The choice of prior is central to solving ill-posed imaging inverse problems, making it essential to select one consistent with the measurements $y$ to avoid severe bias. In Bayesian inverse problems, this could be achieved by evaluating the model evidence $p(y \mid M)$ under different models $M$ th…

• Instance-Wise Adaptive Sampling for Dataset Construction in Approximating Inverse Problem Solutions

  Jiequn Han, Kui Ren, Nathan Soedjak · 20. Februar 2026

  We propose an instance-wise adaptive sampling framework for constructing compact and informative training datasets for supervised learning of inverse problem solutions. Typical learning-based approaches aim to learn a general-purpose inverse map from datasets drawn from a prior distribution, with th…