Related papers: Simple Bounds for Noisy Linear Inverse Problems wi…
In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, "$N$-convex functionals" (the simplest example being the maximum of several fractional-linear functions) of unknown "signal"…
We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of…
Sparse coding and dictionary learning are popular techniques for linear inverse problems such as denoising or inpainting. However in many cases, the measurement process is nonlinear, for example for clipped, quantized or 1-bit measurements.…
We consider a structured estimation problem where an observed matrix is assumed to be generated as an $s$-sparse linear combination of $N$ given $n\times n$ positive-semidefinite matrices. Recovering the unknown $N$-dimensional and…
In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random…
We propose a method to reconstruct sparse signals degraded by a nonlinear distortion and acquired at a limited sampling rate. Our method formulates the reconstruction problem as a nonconvex minimization of the sum of a data fitting term and…
The reconstruction of an unknown quantity from noisy measurements is a mathematical problem relevant in most applied sciences, for example, in medical imaging, radar inverse scattering, or astronomy. This underlying mathematical problem is…
The effectiveness of using model sparsity as a priori information when solving linear inverse problems is studied. We investigate the reconstruction quality of such a method in the non-idealized case and compute some typical recovery errors…
The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank…
Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and…
A sharp phase transition emerges in convex programs when solving the linear inverse problem, which aims to recover a structured signal from its linear measurements. This paper studies this phenomenon in theory under Gaussian random…
We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may…
In presence of sparse noise we propose kernel regression for predicting output vectors which are smooth over a given graph. Sparse noise models the training outputs being corrupted either with missing samples or large perturbations. The…
We investigate the problem of reconstructing n-by-n structured matrix signal X via convex programming, where each column xj is a vector of s-sparsity and all columns have the same l1-norm. The regularizer in use is matrix norm…
We consider the problem of recovering a signal consisting of a superposition of point sources from low-resolution data with a cut-off frequency f. If the distance between the sources is under 1/f, this problem is not well posed in the sense…
We consider the problem of recovering linear image $Bx$ of a signal $x$ known to belong to a given convex compact set ${\cal X}$ from indirect observation $\omega=Ax+\xi$ of $x$ corrupted by random noise $\xi$ with finite covariance matrix.…
We study the problem of learning a sparse linear regression vector under additional conditions on the structure of its sparsity pattern. This problem is relevant in machine learning, statistics and signal processing. It is well known that a…
This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e.g. sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding…
We tackle the problem of recovering an unknown signal observed in an ill-posed inverse problem framework. More precisely, we study a procedure commonly used in numerical analysis or image deblurring: minimizing an empirical loss function…
The observations in many applications consist of counts of discrete events, such as photons hitting a detector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise…