New method for estimating parameters in inverse problems using double robustness.
problem Estimating parameters defined as linear functionals of solutions to linear inverse problems.
method Source condition double robust inference method that uses iterated Tikhonov regularized adversarial estimators.
result Asymptotic normality of the parameter of interest as long as either the primal or dual inverse problem is sufficiently well-posed.
MCGDiff uses SGM to guide SMC for solving ill-posed linear inverse problems.
problem Solving ill-posed linear inverse problems in Bayesian settings.
method Exploiting SGM structure, defining a sequence of intermediate problems, and using SMC methods.
result MCGDiff outperforms competing methods in Bayesian ill-posed inverse problems.
Study solves inverse problems for equations with fractional nonlinearities.
problem Solving inverse problems for semilinear elliptic equations with fractional power nonlinearities.
method Higher order linearization method adapted for fractional order.
result Results of previous studies remain valid for general power nonlinearities.
New method solves linear inverse problems using diffusion models.
problem Linear inverse problems in various domains.
method Posterior sampling with latent diffusion models.
result Provable sample recovery in linear models, outperforming previous methods.
Paper uses SGD for solving linear inverse problems, improving empirical performance.
problem Solving statistical inverse problems in science and engineering.
method Stochastic Gradient Descent (SGD) for linear inverse problems, with smoothing techniques.
result Consistency and finite sample bounds for excess risk demonstrated.
Efficiently solves inverse PDE problems with Gaussian processes.
problem Solving inverse problems in linear PDEs with noisy data.
method Gaussian process regression with algebraic priors.
result High accuracy and computational efficiency achieved.
UCoS avoids forward model evaluations in sampling for large-scale linear inverse problems.
problem Efficient sampling from posterior distributions in large-scale linear inverse problems.
method UCoS approach that learns a task-dependent score function offline and uses affine transformations to derive the conditional score.
result UCoS eliminates the need for forward model evaluations during sampling, making it more efficient.
We solve image inverse problems using a flow-based noise model.
problem Image inverse problems with complex noise patterns.
method Normalizing flow prior for maximum a posteriori estimation.
result Empirical validation on various inverse problems.
New method uses diffusion models for Bayesian inverse problems.
problem Solving Bayesian inverse problems with linear-Gaussian models.
method Decoupled Diffusion Sequential Monte Carlo (DDSMC) method.
result Asymptotically exact solution demonstrated on various data types.
RePS improves diffusion models for solving inverse problems efficiently.
problem Solving inverse problems with incomplete or noisy measurements.
method Restart for Posterior Sampling (RePS) using pre-trained diffusion models.
result RePS achieves faster convergence and superior reconstruction quality.
Paper develops a method to learn optimal sparsity-promoting regularizers for linear inverse problems.
problem Solving linear inverse problems with sparse solutions.
method Bilevel optimization framework to select an optimal synthesis operator B. result Established well-posedness and theoretical guarantees for the learning process.
Variational Gaussian Processes solve linear inverse problems efficiently.
problem Solving inverse problems where indirect observations are corrupted by noise.
method Variational Bayesian methods with Gaussian process priors and inducing variables.
result Posterior contraction rates can be attained by correctly tuned variational procedures.
We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.
The MEM method uses data-driven priors for linear inverse problems, proving convergence and estimating differences.
problem Linear inverse problems with approximate priors.
method Maximum Entropy on the Mean (MEM) method with data-driven priors.
result Empirical mean convergence and estimates for prior differences based on epigraphical distance.
A fast method approximates likelihood scores for noisy linear inverse problems.
problem Solving noisy linear inverse problems efficiently.
method Proposes a simple closed-form approximation to the likelihood score for diffusion and flow-based models.
result Significantly faster than baseline methods while maintaining competitive or better reconstruction performances.
Exact posterior score estimation for solving linear inverse problems
problem Solving linear inverse problems
method Derive the exact posterior score and use it as a denoising training objective
result EPS outperforms training-free and training-based baselines on various metrics
In recent works, both sparsity-based methods as well as learning-based methods have proven to be successful in solving several challenging linear inverse problems. However, sparsity priors for natural signals and images suffer from poor discriminative capability, while learning-based methods seldom provide concrete the…
Deep neural networks solve noisy, complex problems accurately.
problem Reconstructing solutions from noisy, high-dimensional, non-linear inverse problems.
method Restricting infinite-dimensional forward operators to finite-dimensional spaces, training neural networks to approximate these operators robustly to noise.
result Deep neural networks can accurately solve high-dimensional, noisy, non-linear inverse problems.
A new algorithm improves posterior sampling for linear inverse problems.
problem Efficiently sampling from posterior distributions in noisy linear inverse problems.
method Proposes \pddim, a DDIM-type sampler that separately samples along singular directions of the measurement operator.
result The method converges to the Bayesian posterior conditioned on the measurements.
Paper uses RL and diffusion models to solve Bayesian inverse problems.
problem Bayesian inverse problems with latent biases.
method Relative Trajectory Balance (RTB) for RL, conditional diffusion models, off-policy backtracking exploration.
result RTB improves diffusion model posteriors for inverse problems.
New filters improve radar target inference in complex scenarios.
problem Improving radar target inference in highly non-linear system models.
method Developed inverse cubature Kalman filter (I-CKF), inverse quadrature Kalman filter (I-QKF), and inverse cubature-quadrature Kalman filter (I-CQKF) for non-linear systems.
result Numerical experiments show improved estimation accuracy compared to existing methods.
The paper reformulates regression in infinite dimensions as an inverse problem, showing it's equivalent to compact inverse problems.
problem Learning a linear operator between Hilbert spaces from empirical observations.
method Reformulates regression as an inverse problem, proving equivalence to compact inverse problems under specific conditions.
result The inverse problem is equivalent to compact inverse problems in terms of spectral properties and regularisation theory.
Develops inverse EKF for non-linear systems with stability guarantees and learning unknown dynamics.
problem Estimating adversary's Kalman-filtered estimates in highly non-linear systems.
method Proposes inverse extended Kalman filter (I-EKF) for second-order, Gaussian sum, and dithered forward models. Uses reproducing kernel Hilbert space for learning unknown dynamics.
result Derives theoretical stability guarantees for inverse second-order EKF.
We reformulate LIPs as min-max problems for easier solution.
problem Recovering signals from few linear measurements.
method Proposed a min-max reformulation of LIPs.
result Saddle points characterize solutions to LIPs.
In this article we dwell into the class of so called ill posed Linear Inverse Problems (LIP) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from its relatively few random linear measurements. Such problems arise in variety of…
Diffusion models tackle noisy inverse problems with posterior sampling.
problem Efficiently solving general noisy inverse problems.
method Approximation of posterior sampling for diffusion models.
result Diffusion models can handle various noise statistics and nonlinear problems.
New algorithms solve inverse problems using deep learning, converging faster than traditional methods.
problem Solving inverse problems with deep learning models.
method Simple non-convex algorithm for linear and nonlinear inverse problems, with theoretical and empirical support.
result The proposed algorithms converge faster than conventional techniques for certain inverse problems.
This paper tackles regularization parameter learning in inverse problems using data-driven bilevel optimization.
problem Finding optimal regularization parameters in inverse problems.
method Data-driven bilevel optimization approach, analyzing performance in large data samples.
result The approach can reduce computational cost through online numerical schemes based on stochastic gradient descent.
Unified approach for optimizing predictions in linear programming and inverse problems.
problem Optimizing predictions in linear programming and inverse problems.
method Maximum optimality margin approach.
result Unified approach that balances computational efficiency and theoretical properties.
Bayesian inverse problems solved with Gaussian models for PDEs.
problem Solving inverse problems with limited data for PDEs.
method Constructing PDE-informed Gaussian priors for Bayesian inversion.
result PDE-informed Gaussian priors outperform traditional priors.
We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-…
We study two inverse problems on a globally hyperbolic Lorentzian manifold (M,g). The problems are: 1. Passive observations in spacetime: Consider observations in a neighborhood V⊂M of a time-like geodesic μ. Under natural causality conditions, we reconstruct the conformal type of the unknown open, relativ…
Proposes efficient sampling methods for solving linear inverse problems.
problem Solving linear inverse problems with computational efficiency and accuracy.
method Higher-order Langevin diffusion with pre-conditioning and annealing.
result Provable sampling from posterior distributions with accelerated convergence.
Enhances OTA FL algorithms by defining inverse feasibility for linear models.
problem Improving security and privacy in over-the-air federated learning.
method Defines inverse feasibility as an upper bound on condition number, analyzes existing model, proposes new model.
result Proposes a new OTA FL model with enhanced characteristics.
Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well a…
We consider the problem of joint estimation of structured inverse covariance matrices. We perform the estimation using groups of measurements with different covariances of the same unknown structure. Assuming the inverse covariances to span a low dimensional linear subspace in the space of symmetric matrices, our aim i…
We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not…
The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.
problem Inverse problems on holomorphically separable Kähler manifolds and conformally transversally anisotropic manifolds.
method Application of the Stone-Weierstrass theorem to show uniqueness in inverse problems.
result Generalization and simplification of earlier results in inverse problems.
Paper introduces a Gibbs sampler for Bayesian inversion of ill-posed problems.
problem Bayesian inversion of ill-posed problems with linear transformation and additive noise.
method Gibbs algorithm based on prior diffusion model.
result Gibbs algorithm offers a guarantee of convergence in a specific situation.
EnKG solves inverse problems without derivatives, using diffusion models.
problem Solving inverse problems with derivative-free methods.
method Ensemble Kalman Diffusion Guidance (EnKG) using diffusion models.
result EnKG can solve inverse problems with only forward model evaluations.
New method uses deep learning to solve linear inverse problems.
problem Solving linear inverse problems with high-dimensional signals.
method Stochastic coarse-to-fine gradient ascent procedure using implicit prior from denoising CNN.
result General algorithm for solving linear inverse problems without additional training.
We find the optimal Tikhonov regularizer for linear inverse problems without prior knowledge.
problem Finding the optimal regularizer for linear inverse problems in imaging.
method Characterization of the optimal regularizer and learning from data.
result The optimal regularizer is independent of the forward operator and depends only on the mean and covariance of the random variable.
Paper examines convergence rate of PGD for BP objective in inverse problems.
problem Optimizing ill-posed linear inverse problems using BP vs LS.
method Analysis of PGD convergence rate for BP objective, comparison with proximal gradient method.
result PGD converges faster for BP objective due to inherent properties.
Gradient descent and SGD solve nonlinear inverse problems efficiently.
problem Solving nonlinear inverse problems with random design.
method Gradient descent and SGD with mini-batching, under classical assumptions.
result Achieves optimal convergence rates in RKHS framework.
Deep learning has gained great popularity due to its widespread success on many inference problems. We consider the application of deep learning to the sparse linear inverse problem encountered in compressive sensing, where one seeks to recover a sparse signal from a small number of noisy linear measurements. In this p…
Proposes a method to learn both constraints and objective functions from data.
problem Data-driven inverse optimization for mixed-integer linear programs (MILPs).
method Two-stage approach: first learns constraints, then estimates objective-function weights conditioned on learned constraints.
result Proposes and validates a method for learning both objective functions and constraints from data.
A new method for estimating adversarial strategies in nonlinear systems.
problem Inferring an intelligent adversarial agent's strategy in highly nonlinear systems.
method Formulated inverse cognition as a nonlinear Gaussian state-space model and developed an inverse UKF (IUKF) system.
result The estimation error of IUKF converges and closely follows the recursive Cramér-Rao lower bound.
Linear flows on inverse limits of tori are defined and it is shown that two linear flows on an inverse limit of tori are equivalent if and only if there is an automorphism of the inverse limit generating the equivalence.