Inverse Problems

Ask about this topic

Momentum

20 papers in the last four weeks, up 186% on the four weeks before. 0.2% of all new papers.

Jul 13Week of Sep 28
View on map

Latest papers 161

All topics
CardsList
Oct 28, 2025cs.IT

Robustness to Sparse Adversarial Corruption in Arbitrary Linear Measurements: Beyond Exact Recovery

Recovery from linear measurements under sparse adversarial corruption is typically formulated as an exact-recovery problem: one seeks structural conditions on A\mathbf{A} (e.g., restricted isometry property) guaranteeing unique recovery of x⋆\mathbf{x}^\star from y=Ax⋆+e\mathbf{y} = \mathbf{A}\mathbf{x}^\star + \mathbf{e} with ∥e∥0≤q\|\mathbf{e}\|_0 \leq q. However, these guarantees provide no guidance once exact recovery fails. This limitation obscures simple robustness phenomena -- for instance, repeated rows in A\mathbf{A} can preserve nontrivial information about x⋆\mathbf{x}^\star under sparse corruption. In this paper, we study what information about x⋆\mathbf{x}^\star can be \emph{uniformly} recovered from y=Ax⋆+e\mathbf{y} = \mathbf{A}\mathbf{x}^\star + \mathbf{e} for arbitrary A∈Rm×n\mathbf{A}\in\mathbb{R}^{m\times n} and \emph{any} qq-sparse e\mathbf{e}. We show that the robust information is precisely x⋆+ker⁡(U)\mathbf{x}^\star + \ker(\mathbf{U}), where U\mathbf{U} is the orthogonal projection onto the intersection of rowspaces of all submatrices of A\mathbf{A} obtained by deleting 2q2q rows. This clarifies how the row structure of A\mathbf{A} governs whether a qq-sparse corruption allows exact, partial, or only trivial recovery. We further prove every x\mathbf{x} minimizing ∥y−Ax∥0\|\mathbf{y} - \mathbf{A} \mathbf{x}\|_0 belongs to x⋆+ker⁡(U)\mathbf{x}^\star + \ker(\mathbf{U}), yielding a constructive approach to recover this set. For i.i.d. Gaussian matrices, we establish a sharp phase transition between exact and trivial recovery. We sketch two applications: robust network tomography and signal reconstruction from oversampled DCT.

Vishal Halder, Alexandre Reiffers-Masson, Abdeldjalil Aïssa-El-Bey +1

IMT Atlantique, Lab-STICC, CNRS UMR 6285, Brest, France · Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India

Sparse RecoveryAdversarial RobustnessCorruption RobustnessInverse Problems
AskPDF
Oct 6, 2025math.OC

Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions

Data-driven inverse optimization for mixed-integer linear programs (MILPs), which seeks to learn an objective function and constraints consistent with observed decisions, is important for building accurate mathematical models in a variety of domains, including power systems and scheduling. However, to the best of our knowledge, existing data-driven inverse optimization methods primarily focus on learning objective functions under known constraints, and learning both objective functions and constraints from data for MILPs remains largely unexplored. In this paper, we propose a two-stage approach for a class of inverse optimization problems in which the objective is a linear combination of given feature functions and the constraints are parameterized by unknown functions and thresholds. Our method first learns the constraints and then, conditioned on the learned constraints, estimates the objective-function weights. On the theoretical side, we provide finite-sample guarantees for solving the proposed inverse optimization problem. To this end, we develop statistical learning tools for pseudo-metric spaces under sub-Gaussian assumptions and use them to derive a learning-theoretic framework for inverse optimization with both unknown objectives and constraints. On the experimental side, we demonstrate that our method successfully solves inverse optimization problems on scheduling instances formulated as ILPs with up to 100 decision variables.

Akira Kitaoka

Mixed-Integer Linear ProgrammingLearning to OptimizeInverse ProblemsInteger Linear Programming
AskPDF
Oct 2, 2025eess.IV

MACS: Measurement-Aware Consistency Sampling for Inverse Problems

Diffusion models have emerged as powerful generative priors for solving inverse imaging problems. However, their practical deployment is hindered by the substantial computational cost of slow, multi-step sampling. Although Consistency Models (CMs) address this limitation by enabling high-quality generation in only one or a few steps, their direct application to inverse problems has remained largely unexplored. This paper introduces a modified consistency sampling framework specifically designed for inverse problems. The proposed approach regulates the sampler's stochasticity through a measurement-consistency mechanism that leverages the degradation operator, thereby enforcing fidelity to the observed data while preserving the computational efficiency of consistency-based generation. Comprehensive experiments on the Fashion-MNIST and LSUN Bedroom datasets demonstrate consistent improvements across both perceptual and pixel-level metrics, including the Fréchet Inception Distance (FID), Kernel Inception Distance (KID), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM), compared with baseline consistency and diffusion-based sampling methods. The proposed method achieves competitive or superior reconstruction quality with only a small number of sampling steps.

Amirreza Tanevardi, Pooria Abbas Rad Moghadam, Seyed Mohammad Eshtehardian +2

Department of Electrical Engineering Sharif University of Technology Tehran, Iran · Electronics Research Institute Sharif University of Technology Tehran, Iran

Diffusion Model SamplingInverse ProblemsConsistency ModelsImage Inverse Problems
AskPDF
Sep 27, 2025cs.CV

Patch Rebirth: Fast and Transferable Model Inversion of Vision Transformers

Model inversion is a widely adopted technique in data-free learning that reconstructs synthetic inputs from a pretrained model through iterative optimization, without access to original training data. Unfortunately, its application to state-of-the-art Vision Transformers (ViTs) poses a major computational challenge, due to their expensive self-attention mechanisms. To address this, Sparse Model Inversion (SMI) was proposed to improve efficiency by pruning and discarding seemingly unimportant patches, which were even claimed to be obstacles to knowledge transfer. However, our empirical findings suggest the opposite: even randomly selected patches can eventually acquire transferable knowledge through continued inversion. This reveals that discarding any prematurely inverted patches is inefficient, as it suppresses the extraction of class-agnostic features essential for knowledge transfer, along with class-specific features. In this paper, we propose Patch Rebirth Inversion (PRI), a novel approach that incrementally detaches the most important patches during the inversion process to construct sparse synthetic images, while allowing the remaining patches to continue evolving for future selection. This progressive strategy not only improves efficiency, but also encourages initially less informative patches to gradually accumulate more class-relevant knowledge, a phenomenon we refer to as the Re-Birth effect, thereby effectively balancing class-agnostic and class-specific knowledge. Experimental results show that PRI achieves up to 10x faster inversion than standard Dense Model Inversion (DMI) and 2x faster than SMI, while consistently outperforming SMI in accuracy and matching the performance of DMI.

Seongsoo Heo, Dong-Wan Choi

Inha University

Vision TransformerInverse ProblemsData-Efficient Learning
AskPDF
Sep 23, 2025stat.ML

A Gradient Flow Approach to Solving Inverse Problems with Latent Diffusion Models

Solving ill-posed inverse problems requires powerful and flexible priors. We propose leveraging pretrained latent diffusion models for this task through a new training-free approach, termed Diffusion-regularized Wasserstein Gradient Flow (DWGF). Specifically, we formulate the posterior sampling problem as a Wasserstein gradient flow in the latent space of an expected negative log posterior objective, regularized by a Kullback-Leibler divergence to the diffusion prior. We demonstrate the performance of our method on standard benchmarks using StableDiffusion (Rombach et al., 2022) as the prior.

Tim Y. J. Wang, O. Deniz Akyildiz

Department of Mathematics Imperial College London

Wasserstein Gradient FlowsBayesian Inverse ProblemsDiffusion-Based Inverse ProblemsDiffusion Posterior SamplingInverse Problems
AskPDF
May 30, 2025physics.flu-dyn

Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments

Rapid urbanization demands efficient monitoring of turbulent wind and pollutant dispersion, yet existing reconstruction and sensor placement strategies fail under realistic sparsity constraints. Here, we introduce Diff--SPORT, a diffusion-based framework that combines a generative diffusion prior with maximum a posteriori inference and Shapley-value attribution for high-fidelity flow reconstruction and optimal sensor placement. By training a diffusion prior model once over a domain, Diff--SPORT enables non-linear optimal sensor placement and near-real-time flow reconstruction from sparse measurements orders of magnitude faster than RANS or LES simulations, consistently outperforming state-of-the-art methods. The framework also extends, without algorithmic modification, to experimental passive scalar concentration dataset, a direct proxy for pollutant dispersion, measured in a 1:2400 scale water-flume model of the Beijing Haidian neighbourhood under realistic urban flow conditions. Shapley-guided sensor placement achieves up to 57% lower reconstruction error than randomly placed sensors at extreme sparsity, identifying compact and physically interpretable configurations. These results establish Diff--SPORT as a modular foundation offering a zero-shot alternative to retraining-intensive downstream strategies, supporting scalable urban flow monitoring for air quality management and resilient city design.

Abhijeet Vishwasrao, Sai Bharath Chandra Gutha, Andres Cremades +8

Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA · FLOW, Engineering Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden · Division of Robotics, Perception and Learning, School of EECS, KTH Royal Institute of Technology, Stockholm, Sweden +7

Diffusion-Based Inverse ProblemsInverse ProblemsComputational Fluid Dynamics
AskPDF
May 11, 2025cs.LG

Learning from samples: inverse problems over measures

We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO⋆^\star surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.

Francisco Andrade, Gabriel Peyré, Clarice Poon

INRIA (PreMeDICaL & HeKA) · CNRS and ENS, PSL Universit´e · Mathematics Institute, University of Warwick

Inverse Optimal TransportInverse ProblemsOptimal Transport
AskPDF
Jan 25, 2025quant-ph

Superstate Quantum Mechanics

We introduce Superstate Quantum Mechanics (SQM), a theory that considers states in Hilbert space subject to multiple quadratic constraints, with energy'' also expressed as a quadratic function of these states. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization with energy expressed as a quadratic form involving the Hamiltonian. When SQM represents states as unitary operators, the stationary problem becomes a quantum inverse problem with multiple applications in physics, machine learning, and artificial intelligence. Any stationary SQM problem is equivalent to a new algebraic problem that we address in this paper. The non-stationary SQM problem considers the evolution of the system itself, involving the same energy'' operator as in the stationary case. Two possible options for the SQM dynamic equation are considered: (1) within the framework of linear maps from higher-order quantum theory, where 2D-type quantum circuits transform one quantum system into another; and (2) in the form of a Gross-Pitaevskii-type nonlinear map. Although no known physical process currently describes such 2D dynamics, this approach naturally bridges direct and inverse quantum mechanics problems, allowing for the development of a new type of computer algorithms. As an immediately available practical application of the theory, we consider using a quantum channel as a classical computational model; this type of computation can be performed on a classical computer.

Mikhail Gennadievich Belov, Victor Victorovich Dubov, Vadim Konstantinovich Ivanov +3

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, GSP-1, Moscow, Vorob’evy Gory, 119991, Russia · Autretech Group, Skolkovo Innovation Center, Nobel Street, Building 7, Moscow, 121205, Russia · Peter the Great St. Petersburg Polytechnic University, 195251, Russia +1

Inverse Problems
AskPDF
Jul 1, 2024cs.CV

An Expectation-Maximization Algorithm for Training Clean Diffusion Models from Corrupted Observations

Diffusion models excel in solving imaging inverse problems due to their ability to model complex image priors. However, their reliance on large, clean datasets for training limits their practical use where clean data is scarce. In this paper, we propose EMDiffusion, an expectation-maximization (EM) approach to train diffusion models from corrupted observations. Our method alternates between reconstructing clean images from corrupted data using a known diffusion model (E-step) and refining diffusion model weights based on these reconstructions (M-step). This iterative process leads the learned diffusion model to gradually converge to the true clean data distribution. We validate our method through extensive experiments on diverse computational imaging tasks, including random inpainting, denoising, and deblurring, achieving new state-of-the-art performance.

Weimin Bai, Yifei Wang, Wenzheng Chen +1

Academy for Advanced Interdisciplinary Studies, Peking University · College of Future Technology, Peking University · National Biomedical Imaging Center, Peking University +3

Expectation-MaximizationDiffusion-Based Inverse ProblemsInverse ProblemsDiffusion ModelsImage Denoising
AskPDF
Date pendingmath.NA

Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they encode priors and yield efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., plug-and-play methods with learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferentials of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton--Jacobi partial differential equations (HJ PDEs) to develop deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present numerical results in dimensions up to 6464, where the recovered prior is evaluated in a single forward pass.

Oluwatosin Akande, Gabriel P. Langlois, Akwum Onwunta

Industrial and Systems Engineering, Lehigh University, 200 West Packer Avenue, Bethlehem, PA 18015, USA · Department of Mathematics, University of Illinois Urbana-Champaign, Chicago, IL, USA

Inverse Problems
AskPDF
Date pendingcs.LG

Tunable Latent Generative Priors for Compressed Sensing and Inverse Problems

Latent generative models have emerged as powerful priors for solving inverse problems. These models typically represent a class of natural signals at a single, fixed complexity, governed by the latent dimensionality. This can be limiting: depending on the problem, a latent dimensionality that is too small may result in high representation error, while one that is too large may overfit to noise. We develop tunable latent priors for diffusion models, normalizing flows, and variational autoencoders, leveraging nested dropout. Across tasks including compressed sensing, inpainting, denoising, and phase retrieval, we show empirically that tunable priors consistently achieve lower reconstruction errors than fixed-complexity baselines. In the linear denoising setting, we derive the optimal complexity in closed form, showing how it depends on the noise level and the signal spectrum. This work demonstrates the potential of tunable latent generative priors and motivates both the development of supporting theory and their application across a wide range of inverse problems.

Sean Gunn, Jorio Cocola, Oliver De Candido +2

Northeastern University · Harvard University · Technical University of Munich +1

Generative ModelingCompressed SensingInverse ProblemsDiffusion Models
AskPDF