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We consider the bilinear inverse problem of recovering two vectors, $\boldsymbol{x}\in\mathbb{R}^L$ and $\boldsymbol{w}\in\mathbb{R}^L$, from their entrywise product. We consider the case where $\boldsymbol{x}$ and $\boldsymbol{w}$ have…

We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using…

Sparsity is one of the key concepts that allows the recovery of signals that are subsampled at a rate significantly lower than required by the Nyquist-Shannon sampling theorem. Our proposed framework uses arbitrary multiscale transforms,…

Many parameter estimation problems arising in applications are best cast in the framework of Bayesian inversion. This allows not only for an estimate of the parameters, but also for the quantification of uncertainties in the estimates.…

Line spectral estimation theory aims to estimate the off-the-grid spectral components of a time signal with optimal precision. Recent results have shown that it is possible to recover signals having sparse line spectra from few temporal…

In this effort, we propose a convex optimization approach based on weighted $\ell_1$-regularization for reconstructing objects of interest, such as signals or images, that are sparse or compressible in a wavelet basis. We recover the…

In this paper, we study the problem of learning multi-dimensional Gaussian Mixture Models (GMMs), with a specific focus on model order selection and efficient mixing distribution estimation. We first establish an information-theoretic lower…

We consider the problem of recovering a $K$-sparse complex signal $x$ from $m$ intensity measurements. We propose the PhaseCode algorithm, and show that in the noiseless case, PhaseCode can recover an arbitrarily-close-to-one fraction of…

This paper addresses the problem of estimating the modes of an observed non-stationary mixture signal in the presence of an arbitrary distributed noise. A novel Bayesian model is introduced to estimate the model parameters from the…

This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on…

We study unique recovery of cosparse signals from limited-angle tomographic measurements of two- and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality $\ell$ with…

Suppose we are given a vector $f$ in $\R^N$. How many linear measurements do we need to make about $f$ to be able to recover $f$ to within precision $\epsilon$ in the Euclidean ($\ell_2$) metric? Or more exactly, suppose we are interested…

The paper analyses the possibility to recover different biomedical signals if limited number of samples is available. Having in mind that monitoring of health condition is done by measuring and observing key parameters such as heart…

Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem---e.g., in determining the relationship between genetics and the presence or absence of a disease---or they…

We study spectral algorithms in the setting where kernels are learned from data. We introduce the effective span dimension (ESD), an alignment-sensitive complexity measure that depends jointly on the signal, spectrum, and noise level…

We propose and analyze a versatile and efficient multiparameter quantum sensing protocol, which simultaneously estimates many non-commuting and time-dependent signals that are coherently or incoherently coupled to sensing particles. Even in…

We focus in this work on the estimation of the first $k$ eigenvectors of any graph Laplacian using filtering of Gaussian random signals. We prove that we only need $k$ such signals to be able to exactly recover as many of the smallest…

We propose a new regularisation strategy for the classical ensemble Kalman inversion (EKI) framework. The strategy consists of: (i) an adaptive choice for the regularisation parameter in the update formula in EKI, and (ii) criteria for the…

The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data,…

Accurate quantum channel parameter estimation is essential for effective information reconciliation in continuous-variable quantum key distribution (CV-QKD). However, conventional maximum likelihood (ML) estimators rely on a large amount of…