Related papers: Compressed sensing in Hilbert spaces
Compressive sensing is a method to recover the original image from undersampled measurements. In order to overcome the ill-posedness of this inverse problem, image priors are used such as sparsity in the wavelet domain, minimum…
We investigate conditions for the unique recoverability of sparse integer-valued signals from a small number of linear measurements. Both the objective of minimizing the number of nonzero components, the so-called $\ell_0$-norm, as well as…
We prove weak convergence in a separable Hilbert space for estimators of high-dimensional regression coefficients, which yields asymptotic normality and enables direct use of standard asymptotic tools such as the continuous mapping theorem.…
In this paper we look at a well known linear inverse problem that is one of the mathematical cornerstones of the compressed sensing field. In seminal works \cite{CRT,DOnoho06CS} $\ell_1$ optimization and its success when used for recovering…
We consider the problem of recovering a target matrix that is a superposition of low-rank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured high-dimensional signals such…
The linear inverse source and scattering problems are studied from the perspective of compressed sensing, in particular the idea that sufficient incoherence and sparsity guarantee uniqueness of the solution. By introducing the sensor as…
The advancement of sensing technology has driven the widespread application of high-dimensional data. However, issues such as missing entries during acquisition and transmission negatively impact the accuracy of subsequent tasks. Tensor…
This paper will serve as an introduction to the body of work on robust subspace recovery. Robust subspace recovery involves finding an underlying low-dimensional subspace in a dataset that is possibly corrupted with outliers. While this…
We consider the problem of recovering a single or multiple frequency-sparse signals, which share the same frequency components, from a subset of regularly spaced samples. The problem is referred to as continuous compressed sensing (CCS) in…
We study the problem of learning mixtures of low-rank models, i.e. reconstructing multiple low-rank matrices from unlabelled linear measurements of each. This problem enriches two widely studied settings -- low-rank matrix sensing and mixed…
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
This paper studies the problems of identifiability and estimation in high-dimensional nonparametric latent structure models. We introduce an identifiability theorem that generalizes existing conditions, establishing a unified framework…
In sparse recovery, the unique sparsest solution to an under-determined system of linear equations is of main interest. This scheme is commonly proposed to be applied to signal acquisition. In most cases, the signals are not sparse…
In this article we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods. We review the problem of projection-regularization from a statistical point of view. A basic…
We derive an efficient stochastic algorithm for inverse problems that present an unknown linear forcing term and a set of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is…
We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement M and preparation rho the experimenter can determine, given…
Solving an ill-posed linear inverse problem requires knowledge about the underlying signal model. In many applications, this model is a priori unknown and has to be learned from data. However, it is impossible to learn the model using…
In this paper, we study a general framework for compressive sensing assuming the existence of the prior knowledge that $\boldsymbol{x}^*$ belongs to the union of multiple convex sets, $\boldsymbol{x}^{*} \in \bigcup_i \mathcal{C}_i$. In…
We study asymptotically normal estimation and confidence regions for low-dimensional parameters in high-dimensional sparse models. Our approach is based on the $\ell_1$-penalized M-estimator which is used for construction of a bias…
The theory of compressive sensing (CS) asserts that an unknown signal $\mathbf{x} \in \mathbb{C}^N$ can be accurately recovered from $m$ measurements with $m\ll N$ provided that $\mathbf{x}$ is sparse. Most of the recovery algorithms need…