Information theoretic limits for linear prediction with. Information theoretic lower bounds for compressive sensing with generative models. One buzzword you can look up and read more about is the \singlepixel camera. Algorithms and bounds for sensing capacity and compressed sensing with applications to learning graphical models s aeron, m zhao, v saligrama 2008 information theory. These techniques are based on one of the following categories. For example, reference 4 studied the minimum number of noisy measurements required to recover a sparse signal by using shannon information theory bounds. In this paper we introduce a new theory for distributed compressed sensing dcs that enables new distributed. Informationtheoretic bounds of resampling forensics. Aug 17, 2000 detection and recognition problems are modeled as composite hypothesis testing problems involving nuisance parameters. We also propose and prove several interesting statistical properties of the square root of jensenshannon divergence, a wellknown informationtheoretic metric, and exploit other known ones. Spectral compressed sensing via structured matrix completion 1d line spectral estimation as a special case, and indicates how to address multidimensional models. On the other hand, an information theoretic analysis can reveal where there currently exists a gap between the performance of computationally tractable methods, and the fundamental limits.
An informationtheoretic approach to distributed compressed. Tight measurement bounds for exact recovery of structured. Information theoretic lower bounds for compressive sensing with generative models abstract. Introduction sparse vectors are widely used tools in. Consider a population consisting of n individuals, each of whom has one of d types e. Information theoretic bounds for compressed sensing core. In this paper, we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. The goal of compressed sensing is to learn a structured signal x from a limited number of noisy linear measurements y. Zhang jingxiong 1, yangke, guojianzhong2 1school of remote sensing and information engineering, wuhan university, wuhan, china. The problem of sparse estimation via linear measurements commonly referred to as compressive sensing is particularly wellunderstood, with theoretical developments including sharp performance bounds for both practical algorithms 4, 7, 8, 6 and potentially intractable information theoretically optimal algorithms 9, 10, 11, 12. Information theoretic limits for linear prediction with graph.
Information theoretic performance bounds for noisy. I m, where i m is an identity matrix of size m, is assumed to be a. Informationtheoretic bounds on target recognition performance. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by. The course aimed at introducing the topic of compressed sensing cs. Towards an algorithmic theory of compressed sensing, rutgers univ. For comparison, we will use the results by hegde and others 2 in a linear regression setup. Numerical experiments are performed showing the practical use of the technique in signal and image reconstruction from compressed measurements under. Apr 22, 2008 in this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. The focus of our technique is on the replacement of the generalized kullbackleibler divergence, with an information theoretic metric namely the square root of the jensenshannon divergence, which is related to an approximate, symmetrized version of the poisson log likelihood function. Gurumoorthyb, ajit rajwadec, adepartment of electrical engineering, iit bombay binternational center for theoretical sciences, tifr ictstifr, bangalore cdepartment of computer science and engineering, iit bombay abstract. Information theoretic bounds for compressed sensing in sar imaging to cite this article. The standard approach to taking pictures is to rst take a highresolution picture in the \standard basis e.
Information theoretic bounds for compressed sensing article pdf available in ieee transactions on information theory 5610. Tight measurement bounds for exact recovery of structured sparse signals. Index termscompressed sensing, relaxation, fanos method, highdimensional statistical inference, information theoretic bounds, lasso, model selection, signal denoising, sparsity pattern, sparsity recovery, subset selection, support recovery. Informationtheoretic methods in data science edited by. In this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy random projections of data. The obtained bounds establish the relation between the complexity of the autoregressive process and the attainable estimation accuracy through the use of a novel measure of complexity. Compressed sensing cs deals with the reconstruction of sparse signals from a small number of linear measurements. One of the main challenges in cs is to find the support of a sparse signal from a set of noisy observations. Informationtheoretic limits on sparse signal recovery. Reference 5 investigated the contained information in noisy measurements by viewing the measurement system as an information theoretic channel. An informationtheoretic approach to distributed compressed sensing. Compressed mizationsensing bounds prior information weighted n.
Compressed sensing is an emerging field based on the revelation that a small group of linear projections of a sparse signal contains enough information for reconstruction. Bounds for optimal compressed sensing matrices and practical reconstruction schemes shriram sarvotham abstract compressed sensing cs is an emerging. Compressed sensing cs is a new framework for integrated sensing and compression. A strong converse bound for multiple hypothesis testing, with applications to highdimensional estimation. Compressed sensing cs is a new framework for sampling and. Sparsity pattern recovery in compressed sensing by galen reeves a dissertation submitted in partial satisfaction of the requirements for the degree of doctor of philosophy in engineering electrical engineering and computer sciences in the graduate division of the university of california, berkeley committee in charge.
Saligrama, information theoretic bounds to sensing capacity of sensor networks under fixed snr, presented at the information theory workshop, sep. Compressed sensing also known as compressive sensing, compressive sampling, or sparse sampling is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. Since arguments for establishing information theoretic lower bounds are not algorithm speci. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Paper open access related content quantum tomography via. Information theoretic bounds for compressed sensing in sar. We emphasize that although the derivation assumes the measurement matrix to be gaussian, it can be extended to any subgaussian case, by paying a small con. The smaller matrix sa2rm is a compressedd version of the original data a2rn d we start with an overview of di erent constructions of sketching matrices in. We propose a reconstruction algorithm with multiple side in.
Information theoretic bounds for compressed sensing. These problems concern continuous natural phenomena. An interesting question which arises in this context is the e. Finally, we characterize the privacy properties of the compression procedure in informationtheoretic terms, establishing upper bounds on the rate of information communicated between the. The fundamental revelation is that, if an nsample signal x is sparse and has a good kterm approximation in some basis, then it can be reconstructed using m ok lognk n linear projections of x onto another basis.
The fundamental revelation is that, if an n sample signal x is sparse and has a good k term approximation in some basis, then it can be reconstructed using m ok lognk n. Using an information theoretic metric for compressive. Lower bounds for compressed sensing with generative models. On the one hand are rigorous bounds based on information theoretic arguments or the analysis of speci. The problem has received significant interest in compressed sensing and sensor networkssnets literature. Pdf information theoretic bounds for compressed sensing. In this section, we put our work in the context of existing work on poisson compressed sensing with theoretical performance bounds.
Here the authors propose a quantity, named sensing capacity, to incorporate the effects of distortion. On the other hand fundamental information theoretic bounds that are algorithm independent have been presented in 2 1. Informationtheoretic lower bounds for compressive sensing. A novel technique using polar codes signal processing and communications applications conference siu, 2010 ieee 18th,2010 compressed sensing coding and information theory polar codes signal processing. Ieee transactions on information forensics and security 11, 4 2016, 774788. Information theoretic bounds to performance of compressed. In this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. Computer science information theory, computer science machine learning, electrical engineering and systems science signal. In the cs literature, several information theoretic bounds on. Index terms an optimal scaling of the number of observations required forrelaxation, compressed sensing, fanos method, highdimensional statistical inference, information theoretic bounds, sparse approximation, sparse random matrices. Similarly, in 10, the authors consider the matrix completion problem and again use information theoretic techniques to obtain bounds. Detection and information theoretic measures for quantifying the distinguishability between multimedia operator chains.
The improved performance of these methods over their standard counterparts is demonstrated using simulations. Dror baron information theoretic results in compressed sensing compressed sensing. Written by leading experts in a clear, tutorial style, and using consistent notation and definitions throughout, it shows how information theoretic methods are being used in data acquisition, data. On the one hand are rigorous bounds based on informationtheoretic arguments or the analysis of speci. Informationtheoretic limits on sparsity recovery in the. Sep, 20 in the remaining part of this chapter we derive a few information theoretic bounds pertaining to the problem at hand. Compressive sensing provides a new approach to data acquisition and storage. Nowadays, after only 6 years, an abundance of theoretical aspects of compressed sensing are explored in more than articles. Detection and information theoretic measures for quantifying the. It has recently been shown that for compressive sensing, significantly fewer measurements may be required if the sparsity assumption is replaced by the assumption the unknown vector lies near the range of a suitablychosen generative model. In the cs literature, several information theoretic bounds on the scaling law of the required number of measurements for exact support recovery have been. Compressed sensing cs is a new framework for sampling and reconstructing. Furthermore, we show an information theoretic lower bound for tomography of rankr states using adaptive sequences of singlecopy pauli measurements. In this paper we introduce a new theory for distributed compressed sensing dcs that enables new distributed coding algorithms for multisignal.
Informationtheoretic lower bounds for compressive sensing with generative models the goal of standard compressive sensing is to estimate an unknown vecto. Furthermore, x can be reconstructed using linear programming, which has. Using an information theoretic metric for compressive recovery under poisson noise sukanya patila, karthik s. In this paper, we derive some information theory bounds on the performance of noisy compressive sensing to calculate. Compressed regression neural information processing systems. Learn about the stateoftheart at the interface between information theory and data science with this first unified treatment of the subject. Sparse signal recovery with multiple prior information. Moreover, this methodology is to date extensively utilized by applied. The principle observation here is that most natural phenomena of interest is compressible, i. The theory of compressed sensing, where one is interested in recovering a highdimensional signal from a small number of measurements, has grown into a rich field of investigation and found many applications 24. We develop information theoretic performance bounds on target recognition based on statistical models for sensors and data, and examine conditions under which these bounds are tight. Signal processing, compressed sensing, information theory and polar. Bounds for optimal compressed sensing matrices and.
Index terms an optimal scaling of the number of observations required forrelaxation, compressed sensing, fanos method, highdimensional statistical inference, information theoretic bounds. Algorithms and bounds for sensing capacity and compressed sensing with applications to learning graphical models s aeron, m zhao, v saligrama 2008 information theory and applications workshop, 303309, 2008. Introduction to compressed sensing with coding theoretic perspective this book is a course note developed for a graduate level course in spring 2011, at gist, korea. Index termsbasis pursuit, compressed sensing, compressive sampling, informationtheoretic bounds, lasso, orthogonal matching pursuit, prior information, sparsity pattern recovery. Index terms compressive sensing, linear prediction, classi. In the cs literature, several information theoretic bounds on the scaling law of the required number of measurements for exact support recovery have been derived, where. Cs is considered as a new signal acquisition paradigm with which sample taking could be faster than.
We consider two types of distortion for reconstruction. Information theoretic bounds for compressed sensing ieee. Index termsbasis pursuit, compressed sensing, compressive sampling, information theoretic bounds, lasso, orthogonal matching pursuit, prior information, sparsity pattern recovery. Indeed, the informationtheoreticconstrained quadratic programming. Abstract compressed sensing cs deals with the reconstruction of sparse signals from a small number of linear measurements. On the other hand are exact but heuristic predictions made using the replica method from statistical physics. Information theoretic results in compressed sensing. Information theoretic bounds for compressed sensing abstract.125 838 1592 515 760 573 1112 120 1629 850 193 161 87 740 356 1120 216 189 474 762 255 724 1102 1519 714 762 858 1262 1261 858 94 233 831 566 1058