If a signal x is known to have a sparse representation with respect to a frame, the signal can be estimated from a noise-corrupted observation y by finding the best sparse approximation to y. The ability to remove noise in this manner depends on the frame being designed to efficiently represent the signal while it inefficiently represents the noise. This paper analyzes the mean squared error (M...

Abstract—Mixing in the hyperspectral imaging occurs due to the low spatial resolutions of the used cameras. The existing pure materials “endmembers” in the scene share the spectra pixels with different amounts called “abundances”. Unmixing of the data cube is an important task to know the present endmembers in the cube for the analysis of these images. Unsupervised unmixing is done with no info...

We study the problem of estimating large covariance matrices under two types of structural assumptions: (i) The covariance matrix is the summation of a low rank matrix and a sparse matrix, and we have some prior information on the sparsity pattern of the sparse matrix; (ii) The data follow a transelliptical distribution. The former structure regulates the parameter space and has its roots in di...

Standard compressive sensing results state that to exactly recover an s sparse signal in R, one requires O(s · log p) measurements. While this bound is extremely useful in practice, often real world signals are not only sparse, but also exhibit structure in the sparsity pattern. We focus on group-structured patterns in this paper. Under this model, groups of signal coefficients are active (or i...

This paper considers the problem of recovering a one or two dimensional discrete signal which is approximately sparse in its gradient from an incomplete subset of its Fourier coefficients which have been corrupted with noise. We prove that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order s with high probability, it suffices to draw O ...

In this paper, we address the problem of sparse signal recovery from scalar quantized compressed sensing measurements, via optimization. To compensate for compression losses due to dimensionality reduction and quantization, we consider a cost function that is more sparsity-inducing than the commonly used `1-norm. Besides, we enforce a quantization consistency constraint that naturally handles t...

Background Real-time exercise stress cardiac magnetic resonance imaging is challenging due to exaggerated breathing motion and high heart rates; improvements in image reconstruction may help improve the reliability and diagnostic accuracy of this difficult imaging application. Cardiac images possess a rich structure that can be exploited to aid image reconstruction by enforcing sparsity in an a...

Cortical networks are hypothesized to rely on transient network activity to support short-term memory (STM). In this letter, we study the capacity of randomly connected recurrent linear networks for performing STM when the input signals are approximately sparse in some basis. We leverage results from compressed sensing to provide rigorous nonasymptotic recovery guarantees, quantifying the impac...