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Suppose we wish to recover a vector x0 ∈ R (e.g. a digital signal or image) from incomplete and contaminated observations y = Ax0 + e; A is a n by m matrix with far fewer rows than columns (n m) and e is an error term. Is it possible to recover x0 accurately based on the data y? To recover x0, we consider the solution x to the `1-regularization problem min… (More)

We consider the problem of reconstructing a sparse signal x0 ∈ R from a limited number of linear measurements. Given m randomly selected samples of Ux0, where U is an orthonormal matrix, we show that 1 minimization recovers x0 exactly when the number of measurements exceeds m const · μ(U) · S · log n, where S is the number of nonzero components in x0 and μ… (More)

Can we recover a signal f ∈ R from a small number of linear measurements? A series of recent papers developed a collection of results showing that it is surprisingly possible to reconstruct certain types of signals accurately from limited measurements. In a nutshell, suppose that f is compressible in the sense that it is well-approximated by a linear… (More)

- Sohail Bahmani, Justin Romberg
- AISTATS
- 2017

We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with “lifting” and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the… (More)

In the last few years, it has become apparent that traditional wavelet-based image processing algorithms and models have significant shortcomings in their treatment of edge contours. The standard modeling paradigm exploits the fact that wavelet coefficients representing smooth regions in images tend to have small magnitude, and that the multiscale nature of… (More)

This paper considers the problem of estimating the channel response (or Green’s function) between multiple source-receiver pairs. Typically, the channel responses are estimated one-at-a-time: a single source sends out a known probe signal, the receiver measures the probe signal convolved with the channel response, and the responses are recovered using… (More)

- M. Salman Asif, William Mantzel, Justin Romberg
- 2009 47th Annual Allerton Conference on…
- 2009

Blind deconvolution arises naturally when dealing with finite multipath interference on a signal. In this paper we present a new method to protect the signals from the effects of sparse multipath channels — we modulate/encode the signal using random waveforms before transmission and estimate the channel and signal from the observations, without any… (More)

- Ali Ahmed, Felix Krahmer, Justin Romberg
- ArXiv
- 2016

This paper investigates conditions under which certain kinds of systems of bilinear equations have a unique structured solution. In particular, we look at when we can recover vectors w, q from observations of the form yl = 〈w, bl〉〈cl, q〉, l = 1, . . . , L, where bl, cl are known. We show that if w ∈ C M1 and q ∈ C2 are sparse, with no more than K and N… (More)

- Alireza Aghasi, Nam Nguyen, Justin Romberg
- ArXiv
- 2016

Model reduction is a highly desirable process for deep neural networks. While large networks are theoretically capable of learning arbitrarily complex models, overfitting and model redundancy negatively affects the prediction accuracy and model variance. NetTrim is a layer-wise convex framework to prune (sparsify) deep neural networks. The method is… (More)

- M. Salman Asif, Lei Hou Hamilton, Marijn Brummer, Justin Romberg
- Magnetic resonance in medicine
- 2013

Accelerated magnetic resonance imaging techniques reduce signal acquisition time by undersampling k-space. A fundamental problem in accelerated magnetic resonance imaging is the recovery of quality images from undersampled k-space data. Current state-of-the-art recovery algorithms exploit the spatial and temporal structures in underlying images to improve… (More)