Study rates of convergence for approximate solutions to linear ill-posed problems in Hilbert scales.
problem Linear ill-posed inverse problems with noisy data.
method Approximate reconstructions from random noisy data using regularization schemes in Hilbert scale.
result Explicitly established error bounds for smooth regression functions.
In the present paper we consider application of overcomplete dictionaries to solution of general ill-posed linear inverse problems. In the context of regression problems, there has been enormous amount of effort to recover an unknown function using such dictionaries. One of the most popular methods, lasso and its versi…
Novel method uses Gaussian process to estimate particle sizes from scattering data.
problem Estimating particle size distributions from noisy optical scattering measurements.
method Constrained Gaussian process regression with normalization constraints.
result Accurately reconstructs particle size distributions from noisy data.
Paper optimizes estimation of quadratic functionals in nonparametric IV models.
problem Optimal estimation of a nonlinear functional in ill-posed inverse regression.
method Adaptive, minimax estimation using leave-one-out, sieve NPIV estimator with data-driven sieve dimension selection.
result Adaptive estimator achieves minimax optimal rate in various ill-posed cases.
ReTaSA tackles continuous target shift in regression problems.
problem Continuous target shift in regression settings.
method Nonparametric regularized approach to estimate importance weight function.
result The method provides theoretical justification for the estimated importance weight function.
Proposes debiasing strategy for ill-posed regression problems.
problem Estimating functions with conditional moment restrictions, especially when estimators are sensitive to misspecification.
method Debiased estimation using influence function of modified mean squared error.
result Demonstrates finite-sample convergence rate and robustness to misspecification.
Functional PLS improves prediction and inference for scalar responses from functional predictors.
problem Estimating scalar responses from functional predictors in an ill-posed inverse problem.
method Functional partial least squares (PLS) estimator with adaptive early stopping and new tests.
result PLS attains nearly minimax-optimal convergence rates and detects local alternatives.
New method calibrates LV surfaces for exotic derivatives with smoother, more stable Greeks.
problem Challenges in LV calibration leading to spiky surfaces and unstable Greeks.
method Automatic local regression to pre-process market observables and smooth LV surfaces.
result Significantly smoother LV surfaces and greatly improved Greek stability with negligible additional cost.
New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.
problem Comparative analysis of regularization norms in ill-posed problems.
method Small noise analysis framework for Tikhonov and RKHS regularizations.
result Optimal convergence rates achieved with adaptive fractional RKHS, but hyper-parameters decay too fast.
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.
The Fredholm integral equation of the first kind improves solutions for ill-posed supervised learning problems with limited data.
problem Ill-posed supervised learning problems with insufficient data.
method Using the Fredholm integral equation of the first kind (FIFK) with semi-supervised assumptions and MSDF methods.
result Improved accuracy and stability in solutions for ill-posed problems.
This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation, and noisy matrix completion. We propose computationally feasible conv…
Maximum likelihood estimation fails to be well-posed in Gaussian process regression.
problem Establishing well-posedness of maximum likelihood estimation in Gaussian process regression.
method Analyzing the conditions under which maximum likelihood estimation is not Lipschitz in the data with respect to the Hellinger distance.
result Maximum likelihood estimation is not well-posed in the noiseless data setting for any Gaussian process with a stationary covariance function whose lengthscale parameter is estimated using maximum likelihood.
Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner …
IAGAN method improves medical image reconstruction by incorporating adaptive GAN priors.
problem Reconstructing high-fidelity medical images from incomplete data.
method Image-adaptive GAN-based reconstruction method (IAGAN).
result IAGAN can recover fine structures relevant for medical diagnosis.
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.
Researchers develop methods to recover agent behavior from sparse data using Gaussian processes.
problem Recovering agent behavior from limited, noisy data in potential mean field games.
method Two Gaussian process-based frameworks: inf-sup formulation and bilevel approach.
result Surrogate MFG models can accurately reproduce observed data, even when prior information is limited.
A VAE model predicts material properties and microstructures.
problem Building forward and inverse structure-property linkages in materials science.
method Combines VAE with regression, using a two-level prior and multi-modal Gaussian mixture.
result The model achieves accurate forward and inverse predictions of material properties and microstructures.
Score-based models improve diffuse optical tomography accuracy.
problem Improving accuracy in diffuse optical tomography with uncertainty quantification.
method Score-based diffusion models with a mixed score function to prevent overfitting.
result Data-driven prior distribution results in posterior samples with low variance and centred around the ground truth.
Magnetoencephalography and electroencephalography (M/EEG) are non-invasive modalities that measure the weak electromagnetic fields generated by neural activity. Estimating the location and magnitude of the current sources that generated these electromagnetic fields is a challenging ill-posed regression problem known as…
We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spl…
New method for adaptive estimation and inference in econometric models without knowing smoothness.
problem Adaptive estimation and inference in ill-posed linear inverse problems with unknown smoothness.
method Discrepancy principle-based framework for adaptive hyperparameter selection.
result Achieves optimal rates in weak and strong metrics for linear functionals.
We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the con…
Study tackles inverse problems on low-dimensional manifolds, proving stability and proposing a reconstruction algorithm.
problem Inverse problems in infinite-dimensional spaces with nonlinear and ill-posed nature.
method Assumption of low-dimensional manifold, proving stability, proposing Landweber-type algorithm.
result Global convergence of the proposed algorithm, Lipschitz stability for specific inverse problems.
We propose a new learning-based approach to solve ill-posed inverse problems in imaging. We address the case where ground truth training samples are rare and the problem is severely ill-posed - both because of the underlying physics and because we can only get few measurements. This setting is common in geophysical ima…
The paper tackles inverse uncertainty quantification in neutron noise analysis.
problem Uncertainty in estimating material properties from noisy neutron correlation measurements.
method Surrogate models and inverse uncertainty quantification to account for measurement error and model bias.
result Improved prediction of neutron correlations and quantification of uncertainties.
This work tackles uncertainty quantification in tomography reconstruction.
problem Ill-posed nature of tomographic reconstruction leading to no unique solution.
method Gaussian process modeling to incorporate prior knowledge and experimental noises.
result Efficient uncertainty quantification in tomographic reconstruction.
A new mathematical model for the Black-Scholes equation is proposed to forecast option prices. This model includes new interval for the price of the underlying stock as well as new initial and boundary conditions. Conventional notions of maturity time and strike prices are not used. The Black-Scholes equation is solved…
The paper analyzes reg-SGD for convex problems, proving convergence and quantifying the rate of convergence.
problem Minimizing convex, L-smooth functions in a Hilbert space.
method Regularized stochastic gradient descent with decaying regularization.
result Strong convergence to the minimum-norm solution without boundedness assumptions.
We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We …
This paper formulates and studies a general continuous-time behavioral portfolio selection model under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a behavioral model could…
This paper explores deep learning for improving X-ray CT image reconstruction from undersampled data.
problem Improving image reconstruction from undersampled X-ray CT data.
method Analysis of classical and deep learning methods for solving inverse problems.
result Deep learning methods show promise in improving image quality from undersampled data.
Transformer learns context and regularization for ICL in inverse problems.
problem Learning context and effective regularization for transformer-based in-context learning (ICL) in inverse problems.
method Introduced a linear transformer to learn inverse mapping from contextual examples to weight vectors, addressing rank-deficient problems.
result Transformer implicitly learns a prior distribution and effective regularization strategy, outperforming traditional methods.
The paper develops SGD for estimating operators from data.
problem Estimating operators from data in infinite-dimensional spaces.
method Regularized SGD with operator-valued kernels.
result Near-optimal convergence rates for prediction and estimation.
New method forecasts stock option prices accurately.
problem Accurate forecasting of stock option prices.
method Solving the ill-posed Black-Scholes equation using the Quasi-Reversibility Method.
result Good forecasting results demonstrated on market data.
MDNs offer a data-efficient alternative to diffusion and flow models for multimodal scientific learning.
problem Capturing multimodal conditional uncertainty in scientific inverse problems.
method Mixture Density Networks (MDNs) as explicit parametric density estimators.
result MDNs achieve superior generalization, interpretability, and sample efficiency in scientific tasks.
New method learns kernels in nonlocal operators robustly.
problem Learning kernels in nonlocal operators is ill-posed.
method Nonparametric regression with Tikhonov regularization.
result Robust estimator of kernel yields homogenized model.
Learning with noisy labels, which aims to reduce expensive labors on accurate annotations, has become imperative in the Big Data era. Previous noise transition based method has achieved promising results and presented a theoretical guarantee on performance in the case of class-conditional noise. However, this type of a…
New method identifies causal relationships without strong assumptions.
problem Causal Representation Learning (CRL) is ill-posed due to representation and causal discovery issues.
method Identifiability based on grouping of observational variables, self-supervised estimation framework.
result Practical identifiability conditions without temporal structure, interventions, or weak supervision.
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.
Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; however, additional information about the data's structure or on the …
Signal retrieval from a series of indirect measurements is a common task in many imaging, metrology and characterization platforms in science and engineering. Because most of the indirect measurement processes are well-described by physical models, signal retrieval can be solved with an iterative optimization that enfo…
This paper proposes a novel framework to regularize the highly ill-posed and non-linear Fourier ptychography problem using generative models. We demonstrate experimentally that our proposed algorithm, Deep Ptych, outperforms the existing Fourier ptychography techniques, in terms of quality of reconstruction and robustn…
Method determines credit transition matrix from cumulative default probabilities.
problem Quantifying changes in bond credit ratings.
method Setup an ill-posed, linear inverse problem with entropy minimization.
result Method successfully determines CTM from cumulative default probabilities.
New method for inference on strongly identified functionals even when nuisance functions are weakly identified.
problem Inference on continuous linear functionals of weakly identified nuisance functions defined by conditional moment restrictions.
method Proposes penalized minimax estimators for both the primary and debiasing nuisance functions, which can converge to fixed limits regardless of nuisance identifiability.
result Proves the asymptotic normality of a debiased estimator for the functional of interest, leading to asymptotically valid confidence intervals.
Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution ena…
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.
Improved tensor GLM estimation for complex data.
problem Complex tensor data in GLMs leads to high-dimensional, ill-posed estimation.
method Proposed LSRTR-M algorithm using Muon updates for faster convergence and lower errors.
result LSRTR-M converges faster and achieves lower errors than LSRTR.