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We present novel, convex relaxations for rotation and pose estimation problems that can a posteriori guarantee global optimality for practical measurement noise levels. Some such relaxations exist in the literature for specific problem…
Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear…
We study a blind deconvolution problem on graphs, which arises in the context of localizing a few sources that diffuse over networks. While the observations are bilinear functions of the unknown graph filter coefficients and sparse input…
We present a comparison between various algorithms of inference of covariance and precision matrices in small datasets of real vectors, of the typical length and dimension of human brain activity time series retrieved by functional Magnetic…
A new algorithm for the stable solution of a three-dimensional scalar inverse problem of acoustic sounding of an inhomogeneous medium in a cylindrical region is proposed. The data of the problem is the complex amplitude of the wave field,…
We consider an inverse source problem in the stationary radiative transport through an absorbing and scattering medium in two dimensions. Using the angularly resolved radiation measured on an arc of the boundary, we propose a numerical…
A host of problems involve the recovery of structured signals from a dimensionality reduced representation such as a random projection; examples include sparse signals (compressive sensing) and low-rank matrices (matrix completion). Given…
This paper proposes a systematic mathematical analysis of both the direct and inverse acoustic scattering problem given the source in Radon measure space. For the direct problem, we investigate the well-posedness including the existence,…
Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data, whose framework is established by Fefferman et al. \cite{fefferman2020reconstruction,fefferman2021reconstruction}. This paper studies the recovery…
This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of…
We study the Electrical Impedance Tomography Bayesian inverse problem for recovering the conductivity given noisy measurements of the voltage on some boundary surface electrodes. The uncertain conductivity depends linearly on a countable…
Computed ultrasound tomography in echo mode generates maps of tissue speed of sound (SoS) from the shift of echoes when detected under varying steering angles. It solves a linearized inverse problem that requires regularization to…
We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy…
We present a numerical algorithm for nonnegative matrix factorization (NMF) problems under noisy separability. An NMF problem under separability can be stated as one of finding all vertices of the convex hull of data points. The research…
This paper is concerned with the inverse problem to recover a compactly supported Schr{\"o}dinger potential given the differential scattering cross section, i.e. the modulus, but not the phase of the scattering amplitude. To compensate for…
Non-negative matrix factorization (NMF) is a dimensionality reduction technique that has shown promise for analyzing noisy data, especially astronomical data. For these datasets, the observed data may contain negative values due to noise…
Imaging inverse problems aim to recover high-dimensional signals from undersampled, noisy measurements, a fundamentally ill-posed task with infinite solutions in the null-space of the sensing operator. To resolve this ambiguity, prior…
Suppose we wish to recover a signal x in C^n from m intensity measurements of the form |<x,z_i>|^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and…
We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from…
In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if $\Sigma$ is a $2$-dimensional embedded minimal surface, then the knowledge of the…