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Recovering structured models (e.g., sparse or group-sparse vectors, low-rank matrices) given a few linear observations have been well-studied recently. In various applications in signal processing and machine learning, the model of interest is structured in several ways, for example, a matrix that is simultaneously sparse and low rank. Often norms that… (More)

Finding " densely connected clusters " in a graph is in general an important and well studied problem in the literature [12]. It has various applications in pattern recognition, social networking and data mining [11, 14]. Recently, Ames and Vavasis have suggested a novel method for finding cliques in a graph by using convex optimization over the adjacency… (More)

The problem of signal recovery from the autocorrelation, or equivalently, the magnitudes of the Fourier transform, is of paramount importance in various fields of engineering. In this work, for one-dimensional signals, we give conditions, which when satisfied, allow unique recovery from the autocorrelation with very high probability. In particular, for… (More)

Nuclear norm minimization (NNM) has recently gained significant attention for its use in rank minimization problems. Similar to compressed sensing, using null space characterizations, recovery thresholds for NNM have been studied in [12, 4]. However simulations show that the thresholds are far from optimal, especially in the low rank region. In this paper… (More)

Denoising has to do with estimating a signal x0 from its noisy observations y = x0 + z. In this paper, we focus on the " structured denoising problem " , where the signal x0 possesses a certain structure and z has independent normally distributed entries with mean zero and variance σ 2. We employ a structure-inducing convex function f (·) and solve minx{ 1… (More)

Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including ℓ<inf>1</inf> and nuclear norm minimization as well as ℓ<inf>p</inf> minimization with p &#60; 1. These algorithms are known to succeed if certain… (More)

We consider the problem of estimating an unknown but structured signal x<sub>0</sub> from its noisy linear observations y = Ax<sub>0</sub> + z ∈ ℝ<sup>m</sup>. To the structure of x<sub>0</sub> is associated a structure inducing convex function f(·). We assume that the entries of A are i.i.d. standard normal N(0, 1) and z ~ N(0,… (More)

The problem of signal recovery from its Fourier transform magnitude, or equivalently, autocor-relation, is of paramount importance in various fields of engineering and has been around for over 100 years. In order to achieve this, additional structure information about the signal is necessary. In this work, we first provide simple and general conditions,… (More)

Recovering signals from their Fourier transform magnitudes is a classical problem referred to as phase retrieval and has been around for decades. In general, the Fourier transform magnitudes do not carry enough information to uniquely identify the signal and therefore additional prior information is required. In this paper, we shall assume that the… (More)

In this paper we characterize sharp time-data tradeoffs for optimization problems used for solving linear inverse problems. We focus on the minimization of a least-squares objective subject to a constraint defined as the sub-level set of a penalty function. We present a unified convergence analysis of the gradient projection algorithm applied to such… (More)