Related papers: A regularized eigenmatrix method for unstructured …
This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are…
Inverse problems arise in a wide spectrum of applications in fields ranging from engineering to scientific computation. Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, such…
This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors…
We consider the problem of recovering an unknown effectively $(s_1,s_2)$-sparse low-rank-$R$ matrix $X$ with possibly non-orthogonal rank-$1$ decomposition from incomplete and inaccurate linear measurements of the form $y = \mathcal A (X) +…
This note considers the spectral estimation problems of sparse spectral measures under unknown noise levels. The main technical tool is the eigenmatrix method for solving unstructured sparse recovery problems. When the noise level is…
Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the…
In this paper we formulate and solve a robust least squares problem for a system of linear equations subject to quantization error in the data matrix. Ordinary least squares fails to consider uncertainty in the operator, modeling all noise…
We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse regularization techniques that are based on frame representations, we train an encoder-decoder network by including an $\ell^1$-penalty.…
We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under…
Estimating the values of unknown parameters from corrupted measured data faces a lot of challenges in ill-posed problems. In such problems, many fundamental estimation methods fail to provide a meaningful stabilized solution. In this work,…
Modern technologies are producing a wealth of data with complex structures. For instance, in two-dimensional digital imaging, flow cytometry, and electroencephalography, matrix type covariates frequently arise when measurements are obtained…
Recovering a low-complexity signal from its noisy observations by regularization methods is a cornerstone of inverse problems and compressed sensing. Stable recovery ensures that the original signal can be approximated linearly by optimal…
We exploit the similarities between Tikhonov regularization and Bayesian hierarchical models to propose a regularization scheme that acts like a distributed Tikhonov regularization where the amount of regularization varies from component to…
This paper presents an error analysis of classical and learned Tikhonov regularization schemes for inverse problems. We first demonstrate, both theoretically and numerically, that using a fixed regularization parameter across varying noise…
Regularization techniques are necessary to compute meaningful solutions to discrete ill-posed inverse problems. The well-known 2-norm Tikhonov regularization method equipped with a discretization of the gradient operator as regularization…
Sparse recovery is one of the most fundamental and well-studied inverse problems. Standard statistical formulations of the problem are provably solved by general convex programming techniques and more practical, fast (nearly-linear time)…
The problem of numerical differentiation can be thought of as an inverse problem by considering it as solving a Volterra equation. It is well known that such inverse integral problems are ill-posed and one requires regularization methods to…
In this paper we consider the training of single hidden layer neural networks by pseudoinversion, which, in spite of its popularity, is sometimes affected by numerical instability issues. Regularization is known to be effective in such…
An ill-posed inverse problem of autoconvolution type is investigated. This inverse problem occurs in nonlinear optics in the context of ultrashort laser pulse characterization. The novelty of the mathematical model consists in a physically…
Sparse recovery is ubiquitous in machine learning and signal processing. Due to the NP-hard nature of sparse recovery, existing methods are known to suffer either from restrictive (or even unknown) applicability conditions, or high…