International audienceCompressive sensing is a new methodology to cap- ture signals at sub-Nyquist rate. To guarantee exact recovery from compressed measurements, one should choose speciﬁc matrix, which satisﬁes the Restricted Isometry Property (RIP), to implement the sensing procedure. In this letter, we propose to construct the sensing matrix with chaotic sequence following a trivial method and prove that with overwhelming probability, the RIP of this kind of matrix is guaranteed. Meanwhile, its experimental comparisons with Gaussian random matrix, Bernoulli random matrix and sparse matrix are carried out and show that the performances among these sensing matrix are almost equal

Barbot, Jean-Pierre

Sun, Hong

Yu, Lei

Zheng, Gang

HAL-Polytechnique

Compressive Sensing with Chaotic SequenceLei Yu, Jean-Pierre Barbot, Gang Zheng, Hong SunTo cite this version:Lei Yu, Jean-Pierre Barbot, Gang Zheng, Hong Sun. Compressive Sensing with Chaotic Se-quence. IEEE Signal Processing Letters, Institute of Electrical and Electronics Engineers, 2010,17 (8), pp.731 - 734. <inria-00530058>HAL Id: inria-00530058https://hal.inria.fr/inria-00530058Submitted on 29 Oct 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.1Compressive Sensing with Chaotic SequenceLei YU, Jean Pierre BARBOT, Gang ZHENG, and Hong SUN,Abstract—Compressive sensing is a new methodology to cap-ture signals at sub-Nyquist rate. To guarantee exact recoveryfrom compressed measurements, one should choose specificmatrix, which satisfies the Restricted Isometry Property (RIP),to implement the sensing procedure. In this letter, we propose toconstruct the sensing matrix with chaotic sequence following atrivial method and prove that with overwhelming probability,the RIP of this kind of matrix is guaranteed. Meanwhile,its experimental comparisons with Gaussian random matrix,Bernoulli random matrix and sparse matrix are carried outand show that the performances among these sensing matrixare almost equal.Index Terms—Compressive Sensing, Chaos, Logistic Map.I. INTRODUCTIONOver the recent years, a new sampling theory, called Com-pressive Sensing [9], [10], [11] (CS for short), has attractedlots of researchers. The central goal of CS is to captureattributes of a signal using very few measurements: for any N -dimensional signal v (w.l.g. v is s-sparse vector), the measure-ment y 2 RM is captured through v, where s < M < N and 2 RMN is a well chosen matrix satisfying the RestrictedIsometry Property (RIP)[8].Definition 1.1: Matrix  2 Rmn satisfies the RestrictedIsometry Property of order s if there exists a constant  2]0; 1[such that(1  )kvk22  kvk22  (1 + )kvk22 (1)for all s-sparse vectors v.In CS framework, finding a proper sensing matrix satisfying RIP is one of the central problems. Cande`s andTao have proposed that matrix with elements drawn byGaussian distribution or Bernoulli distribution satisfies RIPwith overwhelming probability, providing that sparsity s O(M= logN)[10]. And the randomly selected Fourier basisalso retains RIP with overwhelming probability with sparsitys  O(M=(logN)6)[10]. On the other hand, many researchershave employed some other techniques to construct determin-istic sensing matrix: one group satisfying Statistical IsometryProperty (StRIP) [7], such as Chirp Sensing Codes, secondorder Reed-Muller code, BCH code by R. Calderbank et. al[2], [14], [7]; one group satisfying RIP-1 [5], such as sparserandom matrix by P. Indyk et. al [6] and LDPC by D. BaronThis work was supported by “Bourses Doctorales en Alternance”, PEPS-A2SDC of INSIS-CNRS, NSFC (No. 60872131) and FEDER through CPER2007-2013 with INRIA Lille-Nord Europe.L.Yu and H. Sun are with Signal Processing Laboratory, Electronic andInformation School, Wuhan University, 430079, Wuhan, China. L. Yu and J-P. Barbot are with ECS ENSEA, 6 Avenue du Ponceau, 95014 Cergy-Pontoise,France and EPI ALIEN, INRIA, France. G. Zheng is with INRIA Lille-NordEurope, 40 Avenue Halley, 59650, Villeneuve d’Ascq, France.Email address: yuleiwhu@gmail.com, barbot@ensea.fr,gang.zheng@inria.fr, hongsun@whu.edu.cn.et. al [4]; one group satisfying deterministic RIP, such as finitefields by R. A. Devore [12]; etc.In this paper, we employ chaotic sequence to construct thesensing matrix, called chaotic matrix. Comparing to othertechniques, chaotic system generates the “pseudo-random”matrix in deterministic approach and hence verifies the RIPsimilar to Gaussian or Bernoulli matrix. Moreover, it is easy tobe implemented in physical electric circuit and only one initialstate is necessary to be memorized. Based on the statisticalproperty of chaotic sequence, it is shown that chaotic matrixsatisfies RIP with overwhelming probability, providing thats  O(M= log (N=s)).The main contribution of this paper is to make a con-nection between chaotic sequence and CS. It is shown bythe experiments that the performance of chaotic matrix issomewhat equal to the famous Gaussian random matrix andsparse random matrix. The paper is organized as below. Insection II, one chaotic system is recalled and its statisticalproperty is presented. Section III shows the construction ofchaotic matrix and proves its RIP. In section IV, experimentsare carried out to simulate the performance of chaotic matrix.At the end, the conclusion is given.II. CHAOTIC SEQUENCE AND ITS STATISTICAL PROPERTYLet us consider the following quadratic recurrence equationzn+1 = rzn(1  zn) (2)where r is a positive constant sometimes known as the “bioticpotential” giving the so-called Logistic map. For the specialcase r = 4, the solution for system (2) can be written as below[17]:zn =12[1  cos(22n)] (3)where  2 [0; ] satisfying z0 = 12 [1   cos(2)] with z0the initial condition of (2). It is well known that chaoticsystem (2) can produce very complex sequences. Even more,it is often used as the random number generator in practicesince (2) takes a very simple dynamics [17]. In this section,we will analyze its statistical properties, the distribution, thecorrelations and the sampling distance which guarantees thestatistical independence.Denotexn = cos(22n) (4)obviously, zn takes the similar statistical property with xnsince the linear transformation, for instance the fact that xnand xm are statistically independent, would result in zn andzm are statistically independent.2A. DistributionFunction (4) possesses the following features: zero mean,values bounded within the interval  1  xn  1, invariantdensity given by (x) = 1 (1  x2) 1=2 and x0 = cos(2).B. CorrelationsIt can be checked that the m-th moment of xn satisfiesE(xmn ) = 0 if m is odd andE(xmn ) = 2 mmm2(5)if m is even.C. Statistical independenceIn [19], it has been proved that sequence generated by (4) isnot independent. However, we can measure its independencethrough the high order correlations, which is determined bythe sampling distance. We have the following lemma.Lemma 2.1: Denote X = fxn; xn+1; :::; xn+k; :::g the se-quence generated by (4) with initial state x0 = cos(2), andinteger d the sampling distance, then for any positive integerm0;m1 < 2d, it hasE(xm0n xm1n+d) = E(xm0n )E(xm1n+d): (6)Proof: If there exists at least one odd number in m0;m1,the right side of (6) is equal to 0. For the left side, we haveE(xm0n xm1n+d)=Z 1 1(x0)xm0n xm1n+ddx0=Z 10cosm0(22n) cosm1(22n+d)d=12(m0+m1)X 2nm0Xi=1ni + 2n+dm1Xi=1(n+d)i!where the last equation uses the fact that cos  = (ei+e i)=2andR 10ei2kd = (k), with (k) = 0 if k 6= 0 otherwiseequals to 1.P is the summation over all possible configu-rations, where ni = 1 and (n+d)i = 1.All possible cases for m0 and m1 are analysis as below:1) m1 is odd: jPm0i=1 ni j  m0 and jPm1i=1 (n+d)i j  1,hence 2nPm0i=1 ni + 2n+dPm1i=1 (n+d)i 6= 0;2) m1 is even: it is possible thatPm1i=1 (n+d)i = 0, whilesince m0 is odd (the assumption at the beginning of thispart of proof),Pm0i=1 ni 6= 0, hence 2nPm0i=1 ni +2n+dPm1i=1 (n+d)i 6= 0.Then we can conclude that the left side of (6) is also equal to0.If both m0 and m1 are even numbers, after a trivialcombinatorial analysis, we getE(xm0n xm1n+d) = 2 (m0+m1)m0m0=2m1m1=2Compare it with equation (5), we have (6).(a) (b)(c) (d)Fig. 1. Probability density (xn) (a) and (xn+d) (b); (c) and (d) jointprobability density P (xn; xn+d) for sampling distance d = 5; 15.Remark 2.2: Lemma 2.1 implies that xn and xn+d arestatistically independent when d ! 1, and this result cor-responds to that given in [18]. Approximately, if the sam-pling distance is chosen large enough, for instance d = 15,E(xm0n xm1n+d) = E(xm0n )E(xm1n+d) for all m0;m1 < 32768,hence xn and xn+d can be considered approximately inde-pendent, as illustrated in Fig. 1.III. CHAOTIC SENSING MATRIXLet Z(d; k; z0) = fzn; zn+d; :::; zn+kdg be the chaoticsequence sampled from the output sequence produced byLogistic map (2) with sampling distance d and initial conditionz0, and let xk 2 X(d; k; x0) denote the regularization ofZ(d; k; z0) as belowxk = 1  2zn+kd (7)where X(d; k; x0) just corresponds to equation (4) and hencefulfils the statistical properties discussed in the previous sec-tion.To construct the sensing matrix  2 RMN , generate sam-pled Logistic sequence X(d; k; x0) with length k = M N ,then create a matrix  column by column with this sequence,written as =r2M0BBB@x0 : : : xM(N 1)x1 : : : xM(N 1)+1.........xM 1 : : : xMN 11CCCA (8)where the scalerq2M is for normalization. By chosen sam-pling distance d = 15, then elements of sequence X(d; k; x0)are approximately independent and satisfy identical distribu-tion (x), i.e. a.i.i.d, and hence elements of matrix  area.i.i.d.3Theorem 3.1: Chaotic matrix  2 RMN constructed fol-lowing (8) satisfies RIP for constant  > 0 with overwhelmingprobability, providing that s  O(M= log(N=s)).Remark 3.2: Inherently, this matrix  is sub-gaussian witha.i.i.d elements. In [16], A. Pajor et. al have proved that allsub-gaussian matrix verify the RIP from geometrical point ofview. In what follows, a brief proof following R. Baraniuk’sidea [3] connecting Johnson-Lindenstrauss property [15], [1]and RIP, is presented. Moreover, we can see what Lemma 2.1implies for RIP.Before giving the proof, let us recall a lemma stated in [1].Lemma 3.3: For h 2]0; 1=2[,E[exp (hQ2)] / 1p1  2h; E[Q4] / 3where Q = hx;ui with x being any row vector of  and ubeing any unit vector.Remark 3.4: In Lemma 3.3, / represents approximatelyless, which goes to be strictly  when sampling distanced!1.Proof for Theorem 3.1: The proof contains two parts: firstprove the J-L property for any sub-matrix of , then concludethe RIP using permutation theory.1) J-L property:Denote T the arbitrary column sub matrix of , with indexset jT j = s. For any unit vector u 2 Rs, from Chernoff’sinequality, given some positive value h, it hasPrhkTuk2  1 + i exp ( hM (1 + ))EhexphMkTuk2i exp ( hM (1 + ))  E exp  hQ2M/ exp ( hM (1 + ))1p1  2hM/ exp M2 2=2  3=3= exp( c1()M)where the last inequality is obtained by Taylor expansion andsetting h = 121+ , which is the extremum point, and c1() =2=4  3=6.Similarly, we can calculate the lower bound of its probabil-ity as followsPrhkTuk2  1  i exp (hM (1 + ))Ehexp hMkTuk2i exp (hM (1 + ))  E exp   hQ2M/ exp(hM(1  ))1  h+ 32h2M= exp( c2()M)where the last inequality is obtained by Taylor expansion andsetting h = hopt =  2 +p4+8 523(1 ) , which is the extremumpoint, and c2() = hopt(1  )(1  hopt + 3h2opt=2).Choose c() = minfc1(); c2()g, then one finally getsPrhkTuk2   1  i  2 exp( c()M) (9)2) RIP:For any s-sparse vector v, denote T the set of locations whereelements are nonzero, then jT j =   s  N . The columnsub matrix T defined in previous part can be set up andsatisfies (9). Let us denote E one complementary event ofcondition in (1), i.e.E =nkTuk2   1  o, and denote E the union of all possible complementary events,i.e. E = Ss=1 E. Then one obtainsPr [E ] =[iPr [ETi ] / 2 exp ( c()M)sX=1N 2snsexp ( c()M) 2s(eN=s)s exp ( c()M)= exp (log 2  c ()M + s (log (N=s) + 1) + log s)where, for a fixed constant c3 > 0, whenever s c3M= log(N=s), the bound will only have the exponent withthe exponential   c4M provided that c4  c()   c3[1 +(1+(log s)=s)= logN=s]. Hence we can choose c3 sufficientlysmall to ensure that c4 > 0.Consequently, the probability for satisfying RIP is at least1  Pr [E ] ' 1  2e c4M .IV. EXPERIMENTSAs presented in section III, we choose sampling distanced = 15, then generate the chaotic matrix following (8). Thesynthetic sparse signals v adopted throughout this section arewith only 1 nonzero entries. The locations and signs ofthe peaks are chosen randomly. The measurement vector y iscomputed by y = v. Then the reconstruction v from y issolving by Linear Programming, which is accomplished usingthe SparseLab [13]. The decision for failure reconstruction iskv   vk > 0:1.One interest is the maximum sparsity s which allows exactreconstruction of the signal. The results are given in Fig. 2and show that the maximum sparsity s in the case of chaoticmatrix is similar to that in the case of Sparse matrix [6],Gaussian random matrix and Bernoulli random matrix. Thedelicate experiment for the maximum sparsity s with respectto measurements M for chaotic matrix is given in Fig. 3.Also the probability of successful recovery (recovery rate)for fixed signal size N = 100 and fixed measurement numberM = 50 is compared among these matrices, shown in Fig. 4.The result shows that chaotic matrix performs similar to theother 3 matrices.In addition, to evaluate the influence of the initial conditionof the chaotic system (2), we set the initial state respectivelyto z0 2 f0:1; 0:2; 0:3; 0:4; 0:6; 0:7; 0:8; 0:9g, and redo theexperiment to test the recovery rate, shown in Fig. 5. Theresult shows that the initial state takes no influence to therecovery rate.4Fig. 2. Maximum sparsity for fixed signal size N = 800 and variablenumber of measurements M 2 [100; 500] (left) and for variable signal sizeN 2 [300; 1000] and fixed number of measurements M = 200 (right).Fig. 3. Probability of correct recovery for fixed signal size N = 1000.V. CONCLUSIONIn this paper, we firstly recall the statistical property ofone special chaotic system - Logistic map and prove thatthe generated sequence is approximately independent withsampling distance large enough (for instance d = 15). Thenwe prove that matrix constructed with this sampled chaoticsequence also satisfies RIP with overwhelming probability.From the experiments, it shows that chaotic matrix has thesimilar performance to Sparse matrix, Gaussian random matrixand Bernoulli random matrix.REFERENCES[1] Dimitris Achlioptas. Database-friendly random projections. In Proc.20th ACM SIGMOD-SIGACT-SIGART symposium on Principles ofFig. 4. Recovery rate for different sensing matrix.Fig. 5. Recovery rate for different initial conditions for chaotic sequence.database systems (PODS), pages 274–281, New York, NY, USA, 2001.ACM.[2] L. Applebaum, S. Howard, S. Searle, and R. Calderbank. Chirpsensing codes: Deterministic compressed sensing measurements for fastrecovery. Applied and Computational Harmonic Analysis, 26(2), March2009.[3] Richard Baraniuk, Mark Davenport, Ronald DeVore, and MichaelWakin. A simple proof of the restricted isometry property for randommatrices. Constructive Approximation, 28(3):253–263, December 2008.[4] D. Baron, S. Sarvotham, and R.G. Baraniuk. Bayesian compressivesensing via belief propagation. Signal Processing, IEEE Transactionson, 58(1):269 –280, jan. 2010.[5] R. Berinde, A.C. Gilbert, P. Indyk, H. Karloff, and M.J. Strauss.Combining geometry and combinatorics: A unified approach to sparsesignal recovery. In 46th Annual Allerton Conference on Communication,Control, and Computing,, pages 798 –805, sept. 2008.[6] R. Berinde and P. Indyk. Sparse recovery using sparse random matrices.Technical report, MIT, 2008.[7] R. Calderbank, S. Howard, and S. Jafarpour. Construction of a largeclass of deterministic sensing matrices that satisfy a statistical isometryproperty. Selected Topics in Signal Processing, IEEE Journal of,4(2):358 –374, april 2010.[8] E. J. Cande`s. The restricted isometry property and its implications forcompressed sensing. Compte Rendus de l’Academie des Sciences, SeriesI, 346(9-10), May 2008.[9] E. J. Cande`s and T. Tao. Decoding by linear programming. IEEE Trans.Inform. Theory, 51(12):4203–4215, Dec. 2005.[10] E.J. Cande`s and T. Tao. Near-optimal signal recovery from randomprojections: Universal encoding strategies? Information Theory, IEEETransactions on, 52(12):5406 –5425, dec. 2006.[11] E.J. Cande`s and M.B. Wakin. An introduction to compressive sampling.Signal Processing Magazine, IEEE, 25(2):21 –30, march 2008.[12] R. A. DeVore. Deterministic constructions of compressed sensingmatrices. Journal of Complexity, 23(4-6), August 2007.[13] David Donoho, Iddo Drori, Victoria Stodden, Yaakov Tsaig, and MortezaShahram. Sparselab. http://sparselab.stanford.edu/.[14] S. Howard, R. Calderbank, and S. Searle. A fast reconstruction algorithmfor deterministic compressive sensing using second order reed-mullercodes. Conference on Information Sciences and Systems (CISS), March2008.[15] W. Johnson and J. Lindenstrauss. Extensions of Lipschitz maps into aHilbert space. Contemporary Mathematics, 26:189–206, 1984.[16] Shahar Mendelson, Alain Pajor, and Nicole Tomczak-Jaegermann. Uni-form uncertainty principle for bernoulli and subgaussian ensembles.Constructive Approximation, 28(3):277–289, December 2008.[17] Wolfram Research. ”logistic map”. http://documents.wolfram.com/.[18] Adriana Vlad, Adrian Luca, and Madalin Frunzete. Computational mea-surements of the transient time and of the sampling distance that enablesstatistical independence in the logistic map. In Proc. InternationalConference on Computational Science and Its Applications (ICCSA),pages 703–718, Berlin, Heidelberg, 2009. Springer-Verlag.[19] Kai Wang, Wenjiang Pei, Haishan Xia, Monica Garcia Nustes, and J.A.Gonzalez. Statistical independence in nonlinear maps coupled to non-invertible transformations. Physics Letters A, 372(44):6593 – 6601,2008.

Compressive Sensing with Chaotic Sequence

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HAL Id: inria-00530058https://hal.inria.fr/inria-00530058Submitted on 29 Oct 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Compressive Sensing with Chaotic SequenceLei Yu, Jean-Pierre Barbot, Gang Zheng, Hong SunTo cite this version:Lei Yu, Jean-Pierre Barbot, Gang Zheng, Hong Sun. Compressive Sensing with Chaotic Sequence.IEEE Signal Processing Letters, Institute of Electrical and Electronics Engineers, 2010, 17 (8), pp.731- 734. ￿10.1109/LSP.2010.2052243￿. ￿inria-00530058￿1Compressive Sensing with Chaotic SequenceLei YU, Jean Pierre BARBOT, Gang ZHENG, and Hong SUN,Abstract—Compressive sensing is a new methodology to cap-ture signals at sub-Nyquist rate. To guarantee exact recoveryfrom compressed measurements, one should choose specificmatrix, which satisfies the Restricted Isometry Property (RIP),to implement the sensing procedure. In this letter, we propose toconstruct the sensing matrix with chaotic sequence following atrivial method and prove that with overwhelming probability,the RIP of this kind of matrix is guaranteed. Meanwhile,its experimental comparisons with Gaussian random matrix,Bernoulli random matrix and sparse matrix are carried outand show that the performances among these sensing matrixare almost equal.Index Terms—Compressive Sensing, Chaos, Logistic Map.I. INTRODUCTIONOver the recent years, a new sampling theory, called Com-pressive Sensing [9], [10], [11] (CS for short), has attractedlots of researchers. The central goal of CS is to captureattributes of a signal using very few measurements: for any N -dimensional signal v (w.l.g. v is s-sparse vector), the measure-ment y ∈ RM is captured through Φv, where s < M < N andΦ ∈ RM×N is a well chosen matrix satisfying the RestrictedIsometry Property (RIP)[8].Definition 1.1: Matrix Φ ∈ Rm×n satisfies the RestrictedIsometry Property of order s if there exists a constant δ ∈]0, 1[such that(1− δ)∥v∥22 ≤ ∥Φv∥22 ≤ (1 + δ)∥v∥22 (1)for all s-sparse vectors v.In CS framework, finding a proper sensing matrix Φsatisfying RIP is one of the central problems. Candès andTao have proposed that matrix with elements drawn byGaussian distribution or Bernoulli distribution satisfies RIPwith overwhelming probability, providing that sparsity s ≤O(M/ logN)[10]. And the randomly selected Fourier basisalso retains RIP with overwhelming probability with sparsitys ≤ O(M/(logN)6)[10]. On the other hand, many researchershave employed some other techniques to construct determin-istic sensing matrix: one group satisfying Statistical IsometryProperty (StRIP) [7], such as Chirp Sensing Codes, secondorder Reed-Muller code, BCH code by R. Calderbank et. al[2], [14], [7]; one group satisfying RIP-1 [5], such as sparserandom matrix by P. Indyk et. al [6] and LDPC by D. BaronThis work was supported by “Bourses Doctorales en Alternance”, PEPS-A2SDC of INSIS-CNRS, NSFC (No. 60872131) and FEDER through CPER2007-2013 with INRIA Lille-Nord Europe.L.Yu and H. Sun are with Signal Processing Laboratory, Electronic andInformation School, Wuhan University, 430079, Wuhan, China. L. Yu and J-P. Barbot are with ECS ENSEA, 6 Avenue du Ponceau, 95014 Cergy-Pontoise,France and EPI ALIEN, INRIA, France. G. Zheng is with INRIA Lille-NordEurope, 40 Avenue Halley, 59650, Villeneuve d’Ascq, France.Email address: yuleiwhu@gmail.com, barbot@ensea.fr,gang.zheng@inria.fr, hongsun@whu.edu.cn.et. al [4]; one group satisfying deterministic RIP, such as finitefields by R. A. Devore [12]; etc.In this paper, we employ chaotic sequence to construct thesensing matrix, called chaotic matrix. Comparing to othertechniques, chaotic system generates the “pseudo-random”matrix in deterministic approach and hence verifies the RIPsimilar to Gaussian or Bernoulli matrix. Moreover, it is easy tobe implemented in physical electric circuit and only one initialstate is necessary to be memorized. Based on the statisticalproperty of chaotic sequence, it is shown that chaotic matrixsatisfies RIP with overwhelming probability, providing thats ≤ O(M/ log (N/s)).The main contribution of this paper is to make a con-nection between chaotic sequence and CS. It is shown bythe experiments that the performance of chaotic matrix issomewhat equal to the famous Gaussian random matrix andsparse random matrix. The paper is organized as below. Insection II, one chaotic system is recalled and its statisticalproperty is presented. Section III shows the construction ofchaotic matrix and proves its RIP. In section IV, experimentsare carried out to simulate the performance of chaotic matrix.At the end, the conclusion is given.II. CHAOTIC SEQUENCE AND ITS STATISTICAL PROPERTYLet us consider the following quadratic recurrence equationzn+1 = rzn(1− zn) (2)where r is a positive constant sometimes known as the “bioticpotential” giving the so-called Logistic map. For the specialcase r = 4, the solution for system (2) can be written as below[17]:zn =12[1− cos(2πθ2n)] (3)where θ ∈ [0, π] satisfying z0 = 12 [1 − cos(2πθ)] with z0the initial condition of (2). It is well known that chaoticsystem (2) can produce very complex sequences. Even more,it is often used as the random number generator in practicesince (2) takes a very simple dynamics [17]. In this section,we will analyze its statistical properties, the distribution, thecorrelations and the sampling distance which guarantees thestatistical independence.Denotexn = cos(2πθ2n) (4)obviously, zn takes the similar statistical property with xnsince the linear transformation, for instance the fact that xnand xm are statistically independent, would result in zn andzm are statistically independent.2A. DistributionFunction (4) possesses the following features: zero mean,values bounded within the interval −1 ≤ xn ≤ 1, invariantdensity given by ρ(x) = 1π (1− x2)−1/2 and x0 = cos(2πθ).B. CorrelationsIt can be checked that the m-th moment of xn satisfiesE(xmn ) = 0 if m is odd andE(xmn ) = 2−m(mm2)(5)if m is even.C. Statistical independenceIn [19], it has been proved that sequence generated by (4) isnot independent. However, we can measure its independencethrough the high order correlations, which is determined bythe sampling distance. We have the following lemma.Lemma 2.1: Denote X = {xn, xn+1, ..., xn+k, ...} the se-quence generated by (4) with initial state x0 = cos(2πθ), andinteger d the sampling distance, then for any positive integerm0,m1 < 2d, it hasE(xm0n xm1n+d) = E(xm0n )E(xm1n+d). (6)Proof: If there exists at least one odd number in m0,m1,the right side of (6) is equal to 0. For the left side, we haveE(xm0n xm1n+d)=∫ 1−1ρ(x0)xm0n xm1n+ddx0=∫ 10cosm0(2πθ2n) cosm1(2πθ2n+d)dθ=12(m0+m1)∑σδ(2nm0∑i=1σni + 2n+dm1∑i=1σ(n+d)i)where the last equation uses the fact that cos θ = (eiθ+e−iθ)/2and∫ 10ei2πθkdθ = δ(k), with δ(k) = 0 if k ̸= 0 otherwiseequals to 1.∑σ is the summation over all possible configu-rations, where σni = ±1 and σ(n+d)i = ±1.All possible cases for m0 and m1 are analysis as below:1) m1 is odd: |∑m0i=1 σni | ≤ m0 and |∑m1i=1 σ(n+d)i | ≥ 1,hence 2n∑m0i=1 σni + 2n+d∑m1i=1 σ(n+d)i ̸= 0;2) m1 is even: it is possible that∑m1i=1 σ(n+d)i = 0, whilesince m0 is odd (the assumption at the beginning of thispart of proof),∑m0i=1 σni ̸= 0, hence 2n∑m0i=1 σni +2n+d∑m1i=1 σ(n+d)i ̸= 0.Then we can conclude that the left side of (6) is also equal to0.If both m0 and m1 are even numbers, after a trivialcombinatorial analysis, we getE(xm0n xm1n+d) = 2−(m0+m1)(m0m0/2)(m1m1/2)Compare it with equation (5), we have (6).(a) (b)(c) (d)Fig. 1. Probability density ρ(xn) (a) and ρ(xn+d) (b); (c) and (d) jointprobability density P (xn, xn+d) for sampling distance d = 5, 15.Remark 2.2: Lemma 2.1 implies that xn and xn+d arestatistically independent when d → ∞, and this result cor-responds to that given in [18]. Approximately, if the sam-pling distance is chosen large enough, for instance d = 15,E(xm0n xm1n+d) = E(xm0n )E(xm1n+d) for all m0,m1 < 32768,hence xn and xn+d can be considered approximately inde-pendent, as illustrated in Fig. 1.III. CHAOTIC SENSING MATRIXLet Z(d, k, z0) = {zn, zn+d, ..., zn+kd} be the chaoticsequence sampled from the output sequence produced byLogistic map (2) with sampling distance d and initial conditionz0, and let xk ∈ X(d, k, x0) denote the regularization ofZ(d, k, z0) as belowxk = 1− 2zn+kd (7)where X(d, k, x0) just corresponds to equation (4) and hencefulfils the statistical properties discussed in the previous sec-tion.To construct the sensing matrix Φ ∈ RM×N , generate sam-pled Logistic sequence X(d, k, x0) with length k = M ×N ,then create a matrix Φ column by column with this sequence,written asΦ =√2Mx0 . . . xM(N−1)x1 . . . xM(N−1)+1.........xM−1 . . . xMN−1 (8)where the scaler√2M is for normalization. By chosen sam-pling distance d = 15, then elements of sequence X(d, k, x0)are approximately independent and satisfy identical distribu-tion ρ(x), i.e. a.i.i.d, and hence elements of matrix Φ area.i.i.d.3Theorem 3.1: Chaotic matrix Φ ∈ RM×N constructed fol-lowing (8) satisfies RIP for constant δ > 0 with overwhelmingprobability, providing that s ≤ O(M/ log(N/s)).Remark 3.2: Inherently, this matrix Φ is sub-gaussian witha.i.i.d elements. In [16], A. Pajor et. al have proved that allsub-gaussian matrix verify the RIP from geometrical point ofview. In what follows, a brief proof following R. Baraniuk’sidea [3] connecting Johnson-Lindenstrauss property [15], [1]and RIP, is presented. Moreover, we can see what Lemma 2.1implies for RIP.Before giving the proof, let us recall a lemma stated in [1].Lemma 3.3: For h ∈]0, 1/2[,E[exp (hQ2)] / 1√1− 2h, E[Q4] / 3where Q = ⟨x,u⟩ with x being any row vector of Φ and ubeing any unit vector.Remark 3.4: In Lemma 3.3, / represents approximatelyless, which goes to be strictly ≤ when sampling distanced → ∞.Proof for Theorem 3.1: The proof contains two parts: firstprove the J-L property for any sub-matrix of Φ, then concludethe RIP using permutation theory.1) J-L property:Denote ΦT the arbitrary column sub matrix of Φ, with indexset |T | = s. For any unit vector u ∈ Rs, from Chernoff’sinequality, given some positive value h, it hasPr[∥ΦTu∥2 ≥ 1 + δ]≤ exp (−hM (1 + δ))E[exp(hM∥ΦTu∥2)]≈ exp (−hM (1 + δ))(E[exp(hQ2)])M/ exp (−hM (1 + δ))(1√1− 2h)M/ exp(−M2(δ2/2− δ3/3))= exp(−c1(δ)M)where the last inequality is obtained by Taylor expansion andsetting h = 12δ1+δ , which is the extremum point, and c1(δ) =δ2/4− δ3/6.Similarly, we can calculate the lower bound of its probabil-ity as followsPr[∥ΦTu∥2 ≤ 1− δ]≤ exp (hM (1 + δ))E[exp(−hM∥ΦTu∥2)]≈ exp (hM (1 + δ))(E[exp(−hQ2)])M/ exp(hM(1− δ))(1− h+ 32h2)M= exp(−c2(δ)M)where the last inequality is obtained by Taylor expansion andsetting h = hopt = −2−δ+√4+8δ−5δ23(1−δ) , which is the extremumpoint, and c2(δ) = hopt(1− δ)(1− hopt + 3h2opt/2).Choose c(δ) = min{c1(δ), c2(δ)}, then one finally getsPr[∣∣∣∥ΦTu∥2 − 1∣∣∣ ≥ δ] ≤ 2 exp(−c(δ)M) (9)2) RIP:For any s-sparse vector v, denote T the set of locations whereelements are nonzero, then |T | = κ ≤ s ≪ N . The columnsub matrix ΦT defined in previous part can be set up andsatisfies (9). Let us denote Eκ one complementary event ofcondition in (1), i.e.Eκ ={∣∣∣∥ΦTu∥2 − 1∣∣∣ ≥ δ}, and denote E the union of all possible complementary events,i.e. E =∪sκ=1 Eκ. Then one obtainsPr [E ] =∪iPr [ETi ] / 2 exp (−c(δ)M)s∑κ=1(Nκ)≤ 2s(ns)exp (−c(δ)M)≤ 2s(eN/s)s exp (−c(δ)M)= exp (log 2− c (δ)M + s (log (N/s) + 1) + log s)where, for a fixed constant c3 > 0, whenever s ≤c3M/ log(N/s), the bound will only have the exponent withthe exponential ≤ −c4M provided that c4 ≤ c(δ) − c3[1 +(1+(log s)/s)/ logN/s]. Hence we can choose c3 sufficientlysmall to ensure that c4 > 0.Consequently, the probability for satisfying RIP is at least1− Pr [E ] ' 1− 2e−c4M .IV. EXPERIMENTSAs presented in section III, we choose sampling distanced = 15, then generate the chaotic matrix following (8). Thesynthetic sparse signals v adopted throughout this section arewith only ±1 nonzero entries. The locations and signs ofthe peaks are chosen randomly. The measurement vector y iscomputed by y = Φv. Then the reconstruction v∗ from y issolving by Linear Programming, which is accomplished usingthe SparseLab [13]. The decision for failure reconstruction is∥v∗ − v∥ > 0.1.One interest is the maximum sparsity s which allows exactreconstruction of the signal. The results are given in Fig. 2and show that the maximum sparsity s in the case of chaoticmatrix is similar to that in the case of Sparse matrix [6],Gaussian random matrix and Bernoulli random matrix. Thedelicate experiment for the maximum sparsity s with respectto measurements M for chaotic matrix is given in Fig. 3.Also the probability of successful recovery (recovery rate)for fixed signal size N = 100 and fixed measurement numberM = 50 is compared among these matrices, shown in Fig. 4.The result shows that chaotic matrix performs similar to theother 3 matrices.In addition, to evaluate the influence of the initial conditionof the chaotic system (2), we set the initial state respectivelyto z0 ∈ {0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, 0.9}, and redo theexperiment to test the recovery rate, shown in Fig. 5. Theresult shows that the initial state takes no influence to therecovery rate.4Fig. 2. Maximum sparsity for fixed signal size N = 800 and variablenumber of measurements M ∈ [100, 500] (left) and for variable signal sizeN ∈ [300, 1000] and fixed number of measurements M = 200 (right).Fig. 3. Probability of correct recovery for fixed signal size N = 1000.V. CONCLUSIONIn this paper, we firstly recall the statistical property ofone special chaotic system - Logistic map and prove thatthe generated sequence is approximately independent withsampling distance large enough (for instance d = 15). Thenwe prove that matrix constructed with this sampled chaoticsequence also satisfies RIP with overwhelming probability.From the experiments, it shows that chaotic matrix has thesimilar performance to Sparse matrix, Gaussian random matrixand Bernoulli random matrix.REFERENCES[1] Dimitris Achlioptas. Database-friendly random projections. In Proc.20th ACM SIGMOD-SIGACT-SIGART symposium on Principles ofFig. 4. Recovery rate for different sensing matrix.Fig. 5. Recovery rate for different initial conditions for chaotic sequence.database systems (PODS), pages 274–281, New York, NY, USA, 2001.ACM.[2] L. Applebaum, S. Howard, S. Searle, and R. Calderbank. Chirpsensing codes: Deterministic compressed sensing measurements for fastrecovery. 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Compressive Sensing with Chaotic Sequence

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