Hierarchical isometry properties of hierarchical measurements

Flinth, Axel; Groß, Benedikt; Roth, Ingo; Eisert, Jens; Wunder, Gerhard

doi:10.1016/j.acha.2021.12.006

Cited by 6 publications

(5 citation statements)

References 40 publications

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“…While Theorems 1 and 2 with other results from the literature, e.g. [26], [32], [37], ensure that these sparsity in levels algorithms can recover vectors with structured support, the theory does not provide insight into how the degree of structure or the quality of structure information provided to the algorithms impact performance. The empirical average case performance presented in this section provides such insight to enhance our understanding of the algorithms.…”

Section: Performance Comparisons For Sparsity In Levelsmentioning

confidence: 91%

“…As a result, the theory from standard compressed sensing on sparse vectors and from model based compressed sensing apply directly to the sparsity in levels setting. By exploiting the structure of sparse in levels vectors, algorithms are able to recover vectors with a higher total sparsity using the same number of measurements [26], [32], [37], while the model also provides the theoretical foundation for recovery success in Fourier imaging and other coherent measurement processes [25], [27], [28].…”

Section: Sparsity In Levelsmentioning

confidence: 99%

“…Both of these algorithms 1 have been extended to the sparsity in levels setting [26], [37] by simply utilizing 11), and DetectSupport (k,n) (w l ) in (13). In fact, in [26], 1 The similar Hard Thresholding Pursuit was studied in [32].…”

Section: Greedy Algorithms For Structured Sparsitymentioning

confidence: 99%

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Conjugate Gradient Iterative Hard Thresholding for Structured Sparsity

Blanchard

2022

IEEE Open J. Signal Process.

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Greedy sparse recovery algorithms are studied in the structured sparsity (sparsity in levels) framework. Recovery guarantees are provided for Normalized Iterative Hard Thresholding and Conjugate Gradient Iterative Hard Thresholding in the form of restricted isometry properties for sparsity in levels. Empirical results indicate that CGIHT is comparable to CoSaMP in recovery capability in the structured setting, while maintaining the computational complexity of NIHT. While exploiting structured sparsity improves recovery performance, pessimistic theoretical guarantees mask when practitioners should use these algorithms; the empirical results offer guidance for using the original greedy algorithms over CGIHT in Levels.

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Section: Performance Comparisons For Sparsity In Levelsmentioning

confidence: 91%

Section: Sparsity In Levelsmentioning

confidence: 99%

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Conjugate Gradient Iterative Hard Thresholding for Structured Sparsity

Blanchard

2022

IEEE Open J. Signal Process.

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“…Refs. [16,51] present further examples of the hierarchical measurement framework applied to massive random access without a built-in security scheme.…”

Section: Secure Massive Accessmentioning

confidence: 99%

Hierarchical compressed sensing

Eisert¹,

Flinth²,

Groß³

et al. 2021

Preprint

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Compressed sensing is a paradigm within signal processing that provides the means for recovering structured signals from linear measurements in a highly efficient manner. Originally devised for the recovery of sparse signals, it has become clear that a similar methodology would also carry over to a wealth of other classes of structured signals. In this work, we provide an overview over the theory of compressed sensing for a particularly rich family of such signals, namely those of hierarchically structured signals. Examples of such signals are constituted by blocked vectors, with only few non-vanishing sparse blocks. We present recovery algorithms based on efficient hierarchical hard-thresholding. The algorithms are guaranteed to stable and robustly converge to the correct solution provide the measurement map acts isometrically restricted to the signal class. We then provide a series of results establishing that the required condition for large classes of measurement ensembles. Building upon this machinery, we sketch practical applications of this framework in machine-type and quantum communication.

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“…In the following, we will refrain from presenting any mathematical proofs. These, along with additional results and discussions, can instead found be found in the journal version of this paper, of which a preprint [13] will soon be available on arXiv.…”

Section: Hierarchical Measurement Operatorsmentioning

confidence: 99%

Hierarchical Sparse Recovery from Hierarchically Structured Measurements with Application to Massive Random Access

Gros

Flinth

Roth

et al. 2021

2021 IEEE Statistical Signal Processing Workshop (SSP)

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A new class of measurement operators for structured compressed sensing problems, termed 'hierarchical' measurement operators, is introduced. Standard Kronecker measurement operators are block-oriented treating each block equally whereas the hierarchical measurement operator allows for processing each block with a different matrix, e.g. mixtures of FFT and Gaussian matrices of different column space size and different sample frequencies each, which are received over multiple antennas. We prove that such hierarchical measurement operators exhibit a hierarchical RIP (HiRIP) property provided the involved matrices have a suitable standard RIP, implying recovery guarantees for a class of algorithm involving hierarchical thresholding operations. Thereby, we generalize prior work on the recovery of hierarchically sparse signals from Kroneckerstructured linear measurements. This structure arises in a variety of communication scenarios in the context of massive internet of things. We numerical demonstrate that the fast and scalable HiHTP algorithm is suitable for solving these types of problems and evaluate its performance in terms of sparse signal recovery and block detection capability.

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Hierarchical isometry properties of hierarchical measurements

Cited by 6 publications

References 40 publications

Conjugate Gradient Iterative Hard Thresholding for Structured Sparsity

Conjugate Gradient Iterative Hard Thresholding for Structured Sparsity

Hierarchical compressed sensing

Hierarchical Sparse Recovery from Hierarchically Structured Measurements with Application to Massive Random Access

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