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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 matrices, among others. In various applications in signal processing and machine learning, the model of interest is… (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)

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)

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)

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)

—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 ℓ1 and nuclear norm minimization as well as ℓp minimization with p < 1. These algorithms are known to succeed if certain conditions on the measurement map are satisfied.… (More)

We consider the problem of estimating an unknown signal x 0 from noisy linear observations y = Ax 0 + z ∈ R m. In many practical instances of this problem, x 0 has a certain structure that can be captured by a structure inducing function f (·). For example, 1 norm can be used to encourage a sparse solution. To estimate x 0 with the aid of a convex f (·), we… (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)

—We consider the problem of recovering signals from their power spectral density. This is a classical problem referred to in literature as the phase retrieval problem, and is of paramount importance in many fields of applied sciences. In general, additional prior information about the signal is required to guarantee unique recovery as the mapping from… (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)