Compressive Covariance Sensing: Structure-based compressive sensing beyond sparsity

Romero, Daniel; Ariananda, Dyonisius Dony; Tian, Zhi; Leus, Geert

doi:10.1109/msp.2015.2486805

Cited by 129 publications

(123 citation statements)

References 47 publications

Supporting

Mentioning

122

Contrasting

Order By: Relevance

“…First, in Section 4, we prove the first non-asymptotic analysis of a widely used estimation scheme that reads Θ( √ d) entries per vector according to a sparse ruler [Mof68,RATL16]. Sparse ruler methods are important because, up to constant factors, they minimize ESC among all methods that read a fixed subset of entries from each vector sample.…”

Section: Our Contributions In Briefmentioning

confidence: 99%

“…Work on Toeplitz covariance estimation with the goal of minimizing entry sample complexity has focused on sparse ruler based sampling. This approach has been known for decades [Mof68,PBNH85] and is widely applied in signal processing applications: we refer the reader to [RATL16] for an excellent overview. There has been quite a bit of interest in finding rulers of minimal size -asymptotically, Θ( √ d) is optimal, but in many applications minimizing the leading constant is important [Lee56, Wic63,RSTL88].…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Sample Efficient Toeplitz Covariance Estimation

Eldar¹,

Li²,

Musco³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

We study the sample complexity of estimating the covariance matrix T of a distribution D over d-dimensional vectors, under the assumption that T is Toeplitz. This assumption arises in many signal processing problems, where the covariance between any two measurements only depends on the time or distance between those measurements. 1 We are interested in estimation strategies that may choose to view only a subset of entries in each vector sample x ∼ D, which often equates to reducing hardware and communication requirements in applications ranging from wireless signal processing to advanced imaging. Our goal is to minimize both 1) the number of vector samples drawn from D and 2) the number of entries accessed in each sample.We provide some of the first non-asymptotic bounds on these sample complexity measures that exploit T 's Toeplitz structure, and by doing so, significantly improve on results for generic covariance matrices. These bounds follow from a novel analysis of classical and widely used estimation algorithms (along with some new variants), including methods based on selecting entries from each vector sample according to a so-called sparse ruler.In addition to results that hold for any Toeplitz T , we further study the important setting when T is close to low-rank, which is often the case in practice. We show that methods based on sparse rulers perform even better in this setting, with sample complexity scaling sublinearly in d. Motivated by this finding, we develop a new covariance estimation strategy that further improves on all existing methods in the low-rank case: when T is rank-k or nearly rank-k, it achieves sample complexity depending polynomially on k and only logarithmically on d.Our results utilize tools from random matrix sketching, leverage score based sampling techniques for continuous time signals, and sparse Fourier transform methods. In many cases, we pair our upper bounds with matching or nearly matching lower bounds.

show abstract

Section: Our Contributions In Briefmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Sample Efficient Toeplitz Covariance Estimation

Eldar¹,

Li²,

Musco³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…Sparse rulers have received significant attention in covariance estimation applications [14,[23][24][25][26]. Given a sample x ∼ D with Toeplitz covariance matrix T , if we read the |R| entries of x corresponding to indices in a ruler R, we obtain an estimate of the covariance t s at every distance s. So in principle, with enough samples, x (1) , .…”

Section: Sparse Ruler Based Samplingmentioning

confidence: 99%

Low-Rank Toeplitz Matrix Estimation Via Random Ultra-Sparse Rulers

Lawrence

Musco

et al. 2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

View full text Add to dashboard Cite

We study how to estimate a nearly low-rank Toeplitz covariance matrix T from compressed measurements. Recent work of Qiao and Pal addresses this problem by combining sparse rulers (sparse linear arrays) with frequency finding (sparse Fourier transform) algorithms applied to the Vandermonde decomposition of T . Analytical bounds on the sample complexity are shown, under the assumption of sufficiently large gaps between the frequencies in this decomposition.In this work, we introduce random ultra-sparse rulers and propose an improved approach based on these objects. Our random rulers effectively apply a random permutation to the frequencies in T 's Vandermonde decomposition, letting us avoid frequency gap assumptions and leading to improved sample complexity bounds. In the special case when T is circulant, we theoretically analyze the performance of our method when combined with sparse Fourier transform algorithms based on random hashing. We also show experimentally that our ultra-sparse rulers give significantly more robust and sample efficient estimation then baseline methods.

show abstract

“…Suppose the measurements collected in y are perturbed by zero-mean white Gaussian noise with unit variance, then the least-squares solution has the inverse error covariance or the Fisher information matrix T(L) = E{(g −ĝ)(g −ĝ) H } = Ψ H (L)Ψ(L) that determines the quality of the estimatorsĝ. Therefore, we can use scalar functions of T(L) as a figure of merit to propose the sparse tensor sampling problem (13) where with "optimize" we mean either "maximize" or "minimize" depending on the choice of the scalar function f {·}. Solving (13) is not trivial due to the cardinality constraints.…”

Section: Problem Modelingmentioning

confidence: 99%

Sparse Sampling for Inverse Problems With Tensors

Ortiz-Jiménez

Coutiño

Chepuri

et al. 2019

IEEE Trans. Signal Process.

Self Cite

View full text Add to dashboard Cite

We consider the problem of designing sparse sampling strategies for multidomain signals, which can be represented using tensors that admit a known multilinear decomposition. We leverage the multidomain structure of tensor signals and propose to acquire samples using a Kronecker-structured sensing function, thereby circumventing the curse of dimensionality. For designing such sensing functions, we develop low-complexity greedy algorithms based on submodular optimization methods to compute near-optimal sampling sets. We present several numerical examples, ranging from multi-antenna communications to graph signal processing, to validate the developed theory.

show abstract

Compressive Covariance Sensing: Structure-based compressive sensing beyond sparsity

Cited by 129 publications

References 47 publications

Sample Efficient Toeplitz Covariance Estimation

Sample Efficient Toeplitz Covariance Estimation

Low-Rank Toeplitz Matrix Estimation Via Random Ultra-Sparse Rulers

Sparse Sampling for Inverse Problems With Tensors

Contact Info

Product

Resources

About