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23
Stable Restoration and Separation of Approximately Sparse Signals
"... This paper develops new theory and algorithms to recover signals that are approximately sparse in some general (i.e., basis, frame, overcomplete, or incomplete) dictionary but corrupted by a combination of measurement noise and interference having a sparse representation in a second general diction ..."
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Cited by 16 (9 self)
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This paper develops new theory and algorithms to recover signals that are approximately sparse in some general (i.e., basis, frame, overcomplete, or incomplete) dictionary but corrupted by a combination of measurement noise and interference having a sparse representation in a second general dictionary. Particular applications covered by our framework include the restoration of signals impaired by impulse noise, narrowband interference, or saturation, as well as image inpainting, superresolution, and signal separation. We develop efficient recovery algorithms and deterministic conditions that guarantee stable restoration and separation. Two application examples demonstrate the efficacy of our approach.
Signal Recovery on Incoherent Manifolds
"... Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear submanifold of a highdimensional ambient space. We introduce SPIN, a firstorder projected gradient method to recover the signal com ..."
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Cited by 7 (1 self)
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Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear submanifold of a highdimensional ambient space. We introduce SPIN, a firstorder projected gradient method to recover the signal components. Despite the nonconvex nature of the recovery problem and the possibility of underdetermined measurements, SPIN provably recovers the signal components, provided that the signal manifolds are incoherent and that the measurement operator satisfies a certain restricted isometry property. SPIN significantly extends the scope of current recovery models and algorithms for lowdimensional linear inverse problems and matches (or exceeds) the current state of the art in terms of performance.
Dictionary learning from sparsely corrupted or compressed signals
 In IEEE Intl. Conf. on Acoustics, Speech and Signal Processing
, 2012
"... In this paper, we investigate dictionary learning (DL) from sparsely corrupted or compressed signals. We consider three cases: I) the training signals are corrupted, and the locations of the corruptions are known, II) the locations of the sparse corruptions are unknown, and III) DL from compressed m ..."
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Cited by 5 (2 self)
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In this paper, we investigate dictionary learning (DL) from sparsely corrupted or compressed signals. We consider three cases: I) the training signals are corrupted, and the locations of the corruptions are known, II) the locations of the sparse corruptions are unknown, and III) DL from compressed measurements, as it occurs in blind compressive sensing. We develop two efficient DL algorithms that are capable of learning dictionaries from sparsely corrupted or compressed measurements. Empirical phase transitions and an inpainting example demonstrate the capabilities of our algorithms. Index Terms — Dictionary learning, sparse approximation, compressive sensing, signal restoration, inpainting.
Sparse signal recovery from sparsely corrupted measurements
 in Proc. Inter. Symp. Inf. Theory (ISIT), St. Pertersburg
, 2011
"... Abstract—We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, su ..."
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Abstract—We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, superresolution, signal separation, and the recovery of signals that are corrupted by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a recently developed uncertainty relation and provide corresponding recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge on the support sets of signal and noise. I.
A FactorGraph Approach to Joint OFDM Channel Estimation and Decoding in Impulsive Noise Channels
"... We propose a factorgraphbased approach to joint channel,impulse, symbol and bit estimation (JCISB) of LDPCcoded orthogonal frequency division multiplexing (OFDM) systems in impulsive noise environments. Impulsive noise arises in many modern wireless and wireline communication systems, such as cel ..."
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We propose a factorgraphbased approach to joint channel,impulse, symbol and bit estimation (JCISB) of LDPCcoded orthogonal frequency division multiplexing (OFDM) systems in impulsive noise environments. Impulsive noise arises in many modern wireless and wireline communication systems, such as cellular LTE and powerline communications, due to uncoordinated interference that is much stronger than thermal noise. Our receiver merges prior knowledge of the impulsive noise models with the recently proposed “generalized approximate message passing” (GAMP) algorithm, and softinput softoutput decoding through the sumproduct framework. Unlike the prior work, we explicitly consider channel estimation in the problem formulation. For N subcarriers, the resulting receiver has a complexity of O(N log N), comparable to a typical DFT receiver. Numerical results indicate that the proposed receiver outperforms all prior impulsive noise OFDM decoders with improvements that reach 13dB when compared to the commonly used DFT receiver.
Sparse recovery with coherent tight frame via analysis dantzig selector and analysis lasso,” arXiv:1301.3248
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CoherenceBased Probabilistic Recovery Guarantees for Sparsely Corrupted Signals
"... AbstractIn this paper, we present novel probabilistic recovery guarantees for sparse signals subject to sparse interference, covering varying degrees of knowledge of the signal and in terference support. Our results assume that the sparsifying dictionaries are characterized by coherence parameters ..."
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AbstractIn this paper, we present novel probabilistic recovery guarantees for sparse signals subject to sparse interference, covering varying degrees of knowledge of the signal and in terference support. Our results assume that the sparsifying dictionaries are characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements. I.
1 Lorentzian Iterative Hard Thresholding: Robust Compressed Sensing with Prior Information
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Identification of sparse linear operators
 IEEE Trans. Inf. Theory
"... We consider the problem of identifying a linear deterministic operator from its response to a given probing signal. For a large class of linear operators, we show that stable identifiability is possible if the total support area of the operator’s spreading function satisfies ∆ ≤ 1/2. This result ho ..."
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We consider the problem of identifying a linear deterministic operator from its response to a given probing signal. For a large class of linear operators, we show that stable identifiability is possible if the total support area of the operator’s spreading function satisfies ∆ ≤ 1/2. This result holds for an arbitrary (possibly fragmented) support region of the spreading function, does not impose limitations on the total extent of the support region, and, most importantly, does not require the support region to be known prior to identification. Furthermore, we prove that stable identifiability of almost all operators is possible if ∆ < 1. This result is surprising as it says that there is no penalty for not knowing the support region of the spreading function prior to identification. Algorithms that provably recover all operators with ∆ ≤ 1/2, and almost all operators with ∆ < 1 are presented. 1