Model-based Sketching and Recovery with Expanders

Bah, Bubacarr; Baldassarre, Luca; Cevher, Volkan

doi:10.1137/1.9781611973402.112

Cited by 10 publications

(37 citation statements)

References 34 publications

(130 reference statements)

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“…We retain the complexity of n but got a smaller complexity for d, which is novel. Related results with a similar d were derived in [27,2] but for structure sparse signals in the framework of model-based compressing sensing or sketching. In that framework, one has second order information about x beyond simple sparsity, which is first order information about x.…”

Section: Introductionsupporting

confidence: 56%

On the Construction of Sparse Matrices From Expander Graphs

Bah

Tanner

2018

Front. Appl. Math. Stat.

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We revisit the asymptotic analysis of probabilistic construction of adjacency matrices of expander graphs proposed in [4]. With better bounds we derived a new reduced sample complexity for the number of nonzeros per column of these matrices, precisely d = O (log s (N/s)); as opposed to the standard d = O (log(N/s)). This gives insights into why using small d performed well in numerical experiments involving such matrices. Furthermore, we derive quantitative sampling theorems for our constructions which show our construction outperforming the existing state-of-the-art. We also used our results to compare performance of sparse recovery algorithms where these matrices are used for linear sketching.

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Section: Introductionsupporting

confidence: 56%

On the Construction of Sparse Matrices From Expander Graphs

Bah

Tanner

2018

Front. Appl. Math. Stat.

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“…In Section 6, we show that an approximationtolerant approach similar to AM-IHT succeeds even when the measurement matrix A is itself sparse. Our approach leverages the notion of the restricted isometry property in the ℓ 1 -norm, also called the RIP-1, which was first introduced in [23] and developed further in the model-based context by [27,24,25]. For sparse A, we propose a modification of AM-IHT, which we call AM-IHT with RIP-1.…”

Section: Paper Outlinementioning

confidence: 99%

“…The paper [27] establishes both lower and upper bounds on the number of measurements required to satisfy the model RIP-1 for certain structured sparsity models. Assuming that the measurement matrix A satisfies the model RIP-1, the paper [25] proposes a modification of expander iterative hard thresholding (EIHT) [24], which achieves stable recovery for arbitrary structured sparsity models. As with the other algorithms for model-based compressive sensing, EIHT only works with exact model projection oracles.…”

Section: Prior Workmentioning

confidence: 99%

Approximation Algorithms for Model-Based Compressive Sensing

Hegde

Indyk

Schmidt

2015

IEEE Trans. Inform. Theory

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Compressive Sensing (CS) states that a sparse signal can be recovered from a small number of linear measurements, and that this recovery can be performed efficiently in polynomial time. The framework of model-based compressive sensing (model-CS) leverages additional structure in the signal and provides new recovery schemes that can reduce the number of measurements even further. This idea has led to measurement-efficient recovery schemes for a variety of signal models. However, for any given model, model-CS requires an algorithm that solves the model-projection problem: given a query signal, report the signal in the model that is closest to the query signal. Often, this optimization problem can be computationally very expensive. Moreover, an approximation algorithm is not sufficient to provably recover the signal. As a result, the model-projection problem poses a fundamental obstacle for extending model-CS to many interesting classes of models.In this paper, we introduce a new framework that we call approximation-tolerant modelbased compressive sensing. This framework includes a range of algorithms for sparse recovery that require only approximate solutions for the model-projection problem. In essence, our work removes the aforementioned obstacle to model-based compressive sensing, thereby extending model-CS to a much wider class of signal models. Interestingly, all our algorithms involve both the minimization and maximization variants of the model-projection problem.We instantiate our new framework for a new signal model that we call the Constrained Earth Mover Distance (CEMD) model. This model is particularly useful for signal ensembles where the positions of the nonzero coefficients do not change significantly as a function of spatial (or temporal) location. We develop novel approximation algorithms for both the maximization and the minimization versions of the model-projection problem via graph optimization techniques. Leveraging these algorithms and our framework results in a nearly sample-optimal sparse recovery scheme for the CEMD model.

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“…We call such an approximation G-group-sparse or in short group-sparse. The projection problem is a fundamental step in Model-based Iterative Hard-Thresholding algorithms for solving inverse problems by imposing group structures [7], [19]. More importantly, we seek to also identify the G-groupsupport of the approximationx, that is the G groups that constitute its support.…”

Section: G S ⊆ G |S| ≤ Gmentioning

confidence: 99%

Group-Sparse Model Selection: Hardness and Relaxations

Baldassarre

Bhan

Cevher

et al. 2016

IEEE Trans. Inform. Theory

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Abstract-Group-based sparsity models are instrumental in linear and non-linear regression problems. The main premise of these models is the recovery of "interpretable" signals through the identification of their constituent groups, which can also provably translate in substantial savings in the number of measurements for linear models in compressive sensing. In this paper, we establish a combinatorial framework for group-model selection problems and highlight the underlying tractability issues. In particular, we show that the group-model selection problem is equivalent to the well-known NP-hard weighted maximum coverage problem. Leveraging a graph-based understanding of group models, we describe group structures that enable correct model selection in polynomial time via dynamic programming. Furthermore, we show that popular group structures can be explained by linear inequalities involving totally unimodular matrices, which afford other polynomial time algorithms based on relaxations. We also present a generalization of the group model that allows for within group sparsity, which can be used to model hierarchical sparsity. Finally, we study the Pareto frontier between approximation error and sparsity budget of group-sparse approximations for two tractable models, among which the tree sparsity model, and illustrate selection and computation tradeoffs between our framework and the existing convex relaxations.

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Model-based Sketching and Recovery with Expanders

Cited by 10 publications

References 34 publications

On the Construction of Sparse Matrices From Expander Graphs

On the Construction of Sparse Matrices From Expander Graphs

Approximation Algorithms for Model-Based Compressive Sensing

Group-Sparse Model Selection: Hardness and Relaxations

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