Compressed sensing and best 𝑘-term approximation

Cohen, Albert; Dahmen, Wolfgang; DeVore, Ronald A.

doi:10.1090/s0894-0347-08-00610-3

Cited by 829 publications

(841 citation statements)

References 18 publications

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“…a vector with not more than k non-zero entries. Related models of x were considered in [9], [10], [11], [12], [13] and [14] and extensions to noisy observations were presented in [10], [15] and [16].…”

Section: A Compressed Sensingmentioning

confidence: 99%

See 1 more Smart Citation

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Blumensath¹,

Davies²

2009

IEEE Trans. Inform. Theory

245

361

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Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the number of required samples to certain model parameters.Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically This is a corrected version of the papers in which a few small errors have been corrected. Importantly, the dependence on δ in Theorem 3.3 and its corollaries has been corrected. VERSION: DECEMBER 3, 2009 1 on the number of subspaces in the signal model. In other words, whilst unique linear sampling schemes require a small number of samples depending only on the dimension of the subspaces involved, in order to have stable sampling methods, the number of samples depends necessarily logarithmically on the number of subspaces in the model. These results are then applied to two examples, the standard compressed sensing signal model in which the signal has a sparse representation in an orthonormal basis and to a sparse signal model with additional tree structure.

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Section: A Compressed Sensingmentioning

confidence: 99%

“…This relationship shows that the given error bound is only attainable if the observation dimension depends logarithmically on the signal space dimension. See [11] for details on Gelfand width and the relationship to the restricted isometry condition.…”

Section: A K-sparse Signalsmentioning

confidence: 99%

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Blumensath¹,

Davies²

2009

IEEE Trans. Inform. Theory

245

361

View full text Add to dashboard Cite

show abstract

“…Moreover, the following lemma is well known. Lemma 1.1.1 (See for instance [4]) If A is any m × n matrix and 2k ≤ m, then the following properties are equivalent:…”

Section: The Compressed Sensing Problemmentioning

confidence: 99%

An Alternating $l_1$ Approach to the Compressed Sensing Problem

Chretien¹

2010

IEEE Signal Process. Lett.

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Compressed sensing is a new methodology for constructing sensors which allow sparse signals to be efficiently recovered using only a small number of observations. The recovery problem can often be stated as the one of finding the solution of an underdetermined system of linear equations with the smallest possible support. The most studied relaxation of this hard combinatorial problem is the l1-relaxation consisting of searching for solutions with smallest l1-norm. In this short note, based on the ideas of Lagrangian duality, we introduce an alternating l1 relaxation for the recovery problem enjoying higher recovery rates in practice than the plain l1 relaxation and the recent reweighted l1 method of Candès, Wakin and Boyd.

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“…We will refer to this type of signals as k-approximately sparse signals, or for brevity only approximately sparse signals. More on similar type of problems the interested reader can find in [16].…”

Section: Introductionmentioning

confidence: 99%

“…As expected, β will be a function of C and α. However, β will be an absolute constant independent of n. A similar problem was considered with different proof technique in [16]. In this paper we will provide the explicit values of the allowable constants β.…”

Section: Introductionmentioning

confidence: 99%