Sampling Signals from a Union of Subspaces

Lu, Yue M.; Minh, N.

doi:10.1109/msp.2007.914999

Cited by 54 publications

(39 citation statements)

References 13 publications

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“…The factored representation that we have proposed for modeling sparse signal ensembles is closely related to the recently proposed union-of-subspaces modeling frameworks for CS [25][26][27][28]. What is particularly novel about our treatment is the explicit consideration of the block structure of matrices such as P and Φ, and the explicit accounting for measurement bounds on a sensor-by-sensor basis.…”

Section: Discussionmentioning

confidence: 99%

Measurement Bounds for Sparse Signal Ensembles via Graphical Models

Duarte

Wakin

Baron

et al. 2013

IEEE Trans. Inform. Theory

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Abstract-In compressive sensing, a small collection of linear projections of a sparse signal contains enough information to permit signal recovery. Distributed compressive sensing (DCS) extends this framework by defining ensemble sparsity models, allowing a correlated ensemble of sparse signals to be jointly recovered from a collection of separately acquired compressive measurements. In this paper, we introduce a framework for modeling sparse signal ensembles that quantifies the intra-and inter-signal dependencies within and among the signals. This framework is based on a novel bipartite graph representation that links the sparse signal coefficients with the measurements obtained for each signal. Using our framework, we provide fundamental bounds on the number of noiseless measurements that each sensor must collect to ensure that the signals are jointly recoverable.

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Section: Discussionmentioning

confidence: 99%

Measurement Bounds for Sparse Signal Ensembles via Graphical Models

Duarte

Wakin

Baron

et al. 2013

IEEE Trans. Inform. Theory

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“…The SCS structure is a union of K unidimensional subspaces [4], [3] shared by P channels which leads to an estimation problem exactly and efficiently solvable in the framework of Finite Rate of Innovation sampling [5]. In [3] an estimation algorithm SCS-FRI is proposed and studied.…”

Section: A Problem Definitionmentioning

confidence: 99%

“…A comprehensive analysis was done by Xu [29] for the estimation of covariance matrices in linear array processing. He proposed an OpN 2 q algorithm 4 together with an approximate OpN 2 q estimation of the subspace dimension K. Implementation of the Lanczos algorithm is quite involved in practice and may require costly corrections at each iteration [26], [23], [24], this is why asymptotically more expensive OpN 3 q are generally preferred unless the system is very large. The additional structure on the original data matrix T allows us to lower the complexity from OpN 2 q to OpN log N q, making it appealing even for matrices of modest size, and we will derive a novel criterion to estimate the signal subspace dimension K which requires OpK 2 q computations to be run along the subspace estimation process.…”

Section: Application Of the Fri Approach: Perkmentioning

confidence: 99%

Fast and Robust Parametric Estimation of Jointly Sparse Channels

Barbotin

Vetterli

2012

IEEE J. Emerg. Sel. Topics Circuits Syst.

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If P antennas measure a K-multipath channel with N uniformly sampled measurements per channel, the algorithm possesses an OpKP N log N q complexity and an OpKP N q memory footprint instead of OpP N 3 q and OpP N 2 q for the direct implementation, making it suitable for K ! N . The sparsity is estimated online based on the PER, and the algorithm therefore has a sense of introspection being able to relinquish sparsity if it is lacking.The estimation performances are tested on field measurements with synthetic AWGN, and the proposed algorithm outperforms non-sparse reconstruction in the medium to low SNR range (ď 0dB), increasing the rate of successful symbol decodings by 1{10 th in average, and 1{3 rd in the best case. The experiments also show that the algorithm does not perform worse than a nonsparse estimation algorithm in non-sparse operating conditions, since it may fall-back to it if the PER criterion does not detect a sufficient level of sparsity.The algorithm is also tested against a method assuming a "discrete" sparsity model as in Compressed Sensing (CS). The conducted test indicates a trade-off between speed and accuracy.

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“…This is an instance of a more general uniqueness problem studied in [7,8] for union of subspaces models. In particular [7] shows that M is invertible on the union of subspaces ∪ γ∈Γ S γ if and only if M is invertible on all subspaces S γ + S θ for all γ, θ ∈ Γ. In the context of the cosparse analysis model this yields the following result: Proposition 1.…”

Section: Uniquenessmentioning

confidence: 99%

Cosparse analysis modeling - uniqueness and algorithms

Nam

Davies²,

Elad

et al. 2011

2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In the past decade there has been a great interest in a synthesis-based model for signals, based on sparse and redundant representations. Such a model assumes that the signal of interest can be composed as a linear combination of few columns from a given matrix (the dictionary). An alternative analysis-based model can be envisioned, where an analysis operator multiplies the signal, leading to a cosparse outcome. In this paper, we consider this analysis model, in the context of a generic missing data problem (e.g., compressed sensing, inpainting, source separation, etc.). Our work proposes a uniqueness result for the solution of this problem, based on properties of the analysis operator and the measurement matrix. This paper also considers two pursuit algorithms for solving the missing data problem, an L1-based and a new greedy method. Our simulations demonstrate the appeal of the analysis model, and the success of the pursuit techniques presented.

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Sampling Signals from a Union of Subspaces

Cited by 54 publications

References 13 publications

Measurement Bounds for Sparse Signal Ensembles via Graphical Models

Measurement Bounds for Sparse Signal Ensembles via Graphical Models

Fast and Robust Parametric Estimation of Jointly Sparse Channels

Cosparse analysis modeling - uniqueness and algorithms

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