Related papers: Improved bounds for sparse recovery from subsample…
We initiate the study of trade-offs between sparsity and the number of measurements in sparse recovery schemes for generic norms. Specifically, for a norm $\|\cdot\|$, sparsity parameter $k$, approximation factor $K>0$, and probability of…
Compressive phase retrieval is a popular variant of the standard compressive sensing problem in which the measurements only contain magnitude information. In this paper, motivated by recent advances in deep generative models, we provide…
This paper considers the problem of reconstructing sparse or compressible signals from one-bit quantized measurements. We study a new method that uses a log-sum penalty function, also referred to as the Gaussian entropy, for sparse signal…
In many applications we seek to recover signals from linear measurements far fewer than the ambient dimension, given the signals have exploitable structures such as sparse vectors or low rank matrices. In this paper we work in a general…
In compressed sensing, in order to recover a sparse or nearly sparse vector from possibly noisy measurements, the most popular approach is $\ell_1$-norm minimization. Upper bounds for the $\ell_2$- norm of the error between the true and…
Sparse binary matrices are of great interest in the field of sparse recovery, nonnegative compressed sensing, statistics in networks, and theoretical computer science. This class of matrices makes it possible to perform signal recovery with…
Sparse linear regression with ill-conditioned Gaussian random designs is widely believed to exhibit a statistical/computational gap, but there is surprisingly little formal evidence for this belief, even in the form of examples that are…
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,…
This paper studies sequential methods for recovery of sparse signals in high dimensions. When compared to fixed sample size procedures, in the sparse setting, sequential methods can result in a large reduction in the number of samples…
Is it possible to find the sparsest vector (direction) in a generic subspace $\mathcal{S} \subseteq \mathbb{R}^p$ with $\mathrm{dim}(\mathcal{S})= n < p$? This problem can be considered a homogeneous variant of the sparse recovery problem,…
It has recently been shown that for compressive sensing, significantly fewer measurements may be required if the sparsity assumption is replaced by the assumption the unknown vector lies near the range of a suitably-chosen generative model.…
It is well known that the performance of sparse vector recovery algorithms from compressive measurements can depend on the distribution underlying the non-zero elements of a sparse vector. However, the extent of these effects has yet to be…
We propose a probabilistic framework for interpreting and developing hard thresholding sparse signal reconstruction methods and present several new algorithms based on this framework. The measurements follow an underdetermined linear model,…
We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors ${\boldsymbol{\mathsf{x}}}\in{\mathbb R}^m$ from the noiseless linear measurements…
We consider the problem of estimating the parameters of a linear univariate autoregressive model with sub-Gaussian innovations from a limited sequence of consecutive observations. Assuming that the parameters are compressible, we analyze…
We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of…
We study the robust one-bit compressed sensing problem whose goal is to design an algorithm that faithfully recovers any sparse target vector $\theta_0\in\mathbb{R}^d$ \textit{uniformly} via $m$ quantized noisy measurements. Specifically,…
Weighted $\ell_1$-minimization has been studied as a technique for the reconstruction of a sparse signal from compressively sampled measurements when prior information about the signal, in the form of a support estimate, is available. In…
The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas. In order to derive rigorous recovery results, the measurement map is usually modeled…
Non-convex constraints have recently proven a valuable tool in many optimisation problems. In particular sparsity constraints have had a significant impact on sampling theory, where they are used in Compressed Sensing and allow structured…