New Construction of Deterministic Compressed Sensing Matrices via Singular Linear Spaces over Finite Fields

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1 New Contruction of Determinitic Compreed Sening Matrice via Singular Linear Space over Finite Field XUEMEI LIU College of Science Civil Aviation Univerity of China Tianjin CHINA YINGMO JIE College of Science Civil Aviation Univerity of China Tianjin CHINA Abtract: A an emerging approach of ignal proceing not only ha compreed ening (CS) uccefully compreed ampled ignal with few meaurement but alo ha owned the capabilitie of enuring the eact recovery of ignal However the above-mentioned propertie are baed on the (compreed) ening matrice Hence the contruction of ening matrice i the key problem Compared with the intenive tudy of rom ening matrice only a few determinitic contruction are known In thi paper we provide a family of new contruction of determinitic ening matrice via ingular linear pace over finite field how it better performance than Devore contruction uing polynomial over finite field Key Word: Compreed ening matrice Singular linear pace Coherence Retricted iometry property (RIP) Introduction The traditional Nyuit ampling theorem point out that in order to protect from loing information during ampling ignal we have to ample at leat t- wo time fater than their bwidth However it i too epenive to increae the ampling rate it alo bring complicated iue to our work Therefore it i high time to replace the conventional ampling recontruction operation with lower rate keep the veracity about recovering ignal Meanwhile CS theorem ha uccefully tackled thee problem For a dicrete ignal which can be regarded a a vector in R t with t entrie We want to capture thi ignal with t large by taking a mall number of linear meaurement Each linear meaurement i to calculate the inner product v of with vector v Then the t matri Φ which contain thee vector v i called compreed ening matri the information y Φ which i etracted from by Φ i named the meaurement vector Here arie one uetion: For a given meaurement vector y how can we recontruct the original ignal from y Φ? Even though y Φ i uually ill-poed for < t Donoho Cë 3 make the mot of parity to get that a pare ignal can be recontructed from very few meaurement Thi problem i decribed a finding the paret olution of linear euation y Φ min R t 0 t : Φ y () Thi l 0 -minimization i a combinatorial minimization problem i normally NP-hard 4 In fact thi kind of method 5 ha been ued widely Wherea CS provide a mighty method to recontruct pare ignal with applicable algorithm guarantee the number of meaurement t A ignal i aid to be k-pare if the maimum number of nonzero entrie of it eual to k A we all know that the recontruction of a pare ignal6 play a ignificant in the ignal proceing field Puruing greedy algorithm for l 0 -minimization () i one method to recontruct k-pare ignal There i a famou puruing greedy algorithm called orthogonal matching puruit (OMP) 7 If the number of meaurement Dk log( t δ ) where D i a contant δ (0 036) OMP can recontruct from () with probability urpaing δ However due to the ucce of the recovery proce depending heavily on the property of the ening matri ening matri play an important role in the recovery of ignal There are two kind of ening matrice one i called rom ening matrice whoe entrie are romly drawn from certain probability ditribution which conclude Gauian matrice; Bernoulli matrice; Rom partial orthogonal matrice Another i named determinitic ening matrice whoe propertie are better than rom ening matrice Then another problem emerge: What kind of matrice are appropriate? They mut enure that the prominent information in any k-pare E-ISSN: Volume 5 06

2 or compreible ignal cannot be damaged by the dimenionality reduction from R t down to y R In order to guarantee thi problem Cé Tao have introduced a criterion named retricted i- ometry property (RIP) Definition Let Φ be an t matri If there eit a contant δ k (0 ) uch that for any k-pare ignal R t we have ( δ k ) Φ ( + δ k) () then the matri Φ i aid to atify the RIP of order k the mallet nonnegative number δ k in () i called retricted iometry contant (RIC) of order k Actually uppoe that a ening matri atifie the RIP it RIC i mall enough then OMP can recover pare ignal eactly 3 Adverely if a ening matri doe not meet the RIP we cannot acertain if it could recover the ignal or not There i a relationhip among parity k t Given a k- pare ignal R t which can be accurately recovered from meaurement Then an upper bound of the poible parity i k C/ log(t/) (3) where C i a contant 4 Thank to the romne of entrie of rom ening matrice the upper bound of k in (3) ha been achieved by rom ening matrice which could recover pare ignal with high probability8 Wherea there are alo ome d- eficiencie about rom ening matrice Firt rom ening matrice need a lot of torage pace to tore their entrie Second there i no efficient algorithm teting whether a rom ening matri could atify the RIP or not let alone with high probability But the determinitic ening matrice conuer thoe deficiencie Definition 5 Let Φ be a matri with column u u u t the coherence of Φ i defined a µ(φ) ma b j u b u j u b u j for b j t (4) A a coherence can be looked a an euivalent form of the RIP it i an important iue in the determinitic contruction Lemma 3 6 Suppoe Φ i a matri with coherence µ Then Φ atifie the RIP of order k with δ k µ(k ) whenever k < µ + For an t matri Φ there i a famou Welch bound 7 t µ(φ) (t ) (5) which mean that the determinitic contruction depended on coherence can only obtain ening matrice with the RIP of order k O( / ) Briefly There are two tep about CS theory In the firt tep we deign a CS matri Φ that enure that the alient information in any k-pare or compreible ignal i not damaged by the dimenionality reduction from R t down to y R In the econd tep we develop a recontruction algorithm to recover from the meaurement y Here we put our focu on the firt tep Recently ome determinitic contruction of ening matrice have been preented Devore polynomial over finite field8; Gao F algebraic curve 9; Amini Marvati bipolar matri by BCH code 0 it generalization ; Bourgain additive combinatoric6 In thi paper we contruct one kind of determinitic contruction baed on ingular linear pace over finite field how it ecellent propertie Singular linear pace In thi ection we hall introduce the concept of ubpace of type (m h) in ingular linear pace (ee Wang et al ) provide everal lemma Let F be a finite field with element where i a prime power For two non-negative integer n l F (n+l) denote the (n+l)-dimenional row vector pace over F The et of all (n+l) (n+l) noningular matrice over F of the form ( T T 0 T ) where T T are noningular n n l l matrice repectively form a group under matri multiplication called the ingular general linear group of degree n + l over F denoted by GL n+ln (F ) If l 0 (rep n 0) GL nn (F ) GL n (F ) (rep GL l0 (F ) GL l (F )) i the general linear group of degree n (rep l) (See Wan 3) Let P be a m-dimenional ubpace of F (n+l) denote alo by P an m (n+l) matri of rank m whoe row pan the ubpace P call the matri P a matri repreentation of the ubpace P There i an action of GL n+ln (F ) on F (n+l) defined a follow F (n+l) GL n+ln (F ) F (n+l) (( n n+ n+l ) T ) ( n n+ n+l )T The above action induce an action on the et of ubpace of F (n+l) ie a ubpace P i carried by E-ISSN: Volume 5 06

3 T GL n+ln (F ) to the ubpace P T The vector pace F (n+l) together with the above group action i called the (n + l)-dimenional ingular linear pace over F For j n + l let e j be the row vector in whoe j-th coordinate i all other coordinate are 0 Denote by E the l-dimenional ubpace of F (n+l) generated by e n+ e n+ e n+l A m- F (n+l) dimenional ubpace P of F (n+l) i called a ubpace of type (m h) if dim(p E) h The collection of all the ubpace of type (m 0) in F (n+l) where 0 m n i the attenuated pace (ee AE Brouwer et al 4 Lemma 4 Let V denote the (n + l)-dimenional row vector pace over a finite field F fi a ubpace W of type (n + l d h) contained in V Let M + (i ; d h; n + l n) denote the et of all ubpace U of type (i ) contained in V atifying U + W V let N + (i ; d h; n + l n) denote the ize of M + (i ; d h; n + l n) Then N + (i ; d h; n + l n) d(n+l i ) n + l d h h i d (6) Proof By the tranitivity of GL n+ln (F ) on the et of ubpace of the ame type we may chooe the ubpace W of type (n + l d h) a the form ( I (n+l d h) 0 (n+l d hd+h l) I (h) 0 (hl h) Let U ha a matri repreentation of the form ( X (i d n+l d h) 0 (i d d+h l) 0 0 Y (dn+l d h) I (d+h l) B (dh) I (dl h) 0 ( n+l d h) 0 A ( h) 0 ( l h) ) where X i an (i d ) (n + l d h) matri of rank (i d ) Y i a d (n+l d h) matri A i an h matri of rank B i a d h matri Then X i an (i d )-ubpace which contained in I (n+l d h) By Wan (3 00b Theorem n + l d h 7) there are choice for X By i d the ame token A i an -ubpace which contained h in I (h) ha choice By the tranitivity of GL n+ln (F ) we may let X (I (i d ) A (I () 0 (i d n+l h i + ) ) 0 (h ) ) ) Then U ha the uniue matri repreentation of the form ( ) I (α) 0 (αβ) 0 (αγ) 0 (α ) Y (dβ) 0 (dγ) 0 (d ) B (dh ) 0 (dl h) 0 ( α) 0 0 ( γ) I ( ) 0 0 ( l h) where α i d β n + l h i + γ h + d l Hence N + (i ; d h; n + l n) d(n+l i ) n + l d h h i d h Lemma 5 Let V denote the (n + l)-dimenional row vector pace over a finite field F fi a ubpace W of type (n + l d h) contained in V For a given ubpace U of type (i h ) contained in V atifying U + W V let u(n + l d h; i ; i h ) denote the number of ubpace U of type (i ) contained in V atifying U + W V U U Then u(n + l d h; i ; i h ) d(i i ) i d h h i d (7) Proof Since the ubgroup GL n+ln (F ) W of GL n+ln (F ) fiing W act tranitively on the et {U U + W V dimu i } the number u(n + l d h; i ; i h ) depend only on i i By Lemma (6) we get u(n + l d h; i ; i h ) d(i i ) i d h i d h Lemma 6 Let V denote the (n + l)-dimenional row vector pace over a finite field F fi a ubpace W of type (n + l d h) contained in V For a given ubpace U of type (i ) contained in V atifying U + W V let u (n + l d h; i ; i h ) denote the number of ubpace U of type (i h ) contained in V atifying U + W V U U Then u (n + l d h; i ; i h ) n + l h i + h h i h i + h (8) E-ISSN: Volume 5 06

4 Proof Let M (U U ) U M + (i ; d h; n + l n) U M + (i h ; d h; n + l n) U U We compute the ize of M in the following two way For a fied ubpace U of type (i ) there are u (n + l d h; i ; i h ) ubpace of type (i h ) containing U By Lemma 4 M u (n + l d h; i ; i h ) N + (i ; d h; n + l n) (9) For a fied ubpace U of type (i h ) there are u(n + l d h; i ; i h ) ubpace of type (i ) contained in U By Lemma 4 M u(n + l d h; i ; i h ) N + (i h ; d h; n + l n) (0) Combining (9) (0) (7) (6) (8) hold Lemma 7 Given integer 0 h l d i n + l h n + d the euence N + (i ; d h; n + l n) i unimodal get it peak at i n+l h + Proof By Lemma 4 if i < i then we have N + (i ; d h; n + l n) N + (i ; d h; n + l n) d(n+l i ) n + l d h h i d h d(n+l i ) n + l d h h i d d(i i ) where i h d ii d+ n+l h i + in+l h i + + ( i ) ( i ) ( i + d ) ( i d ) ( n+l h i + + ) ( n+l h i + ) i + d i d n+l h i + n+l h i + + i + d i + d n+l h i + < n+l h i + i d < < n+l h i + + If i n+l h + then i < n + l h i + + Hence when we have i d n+l h i + + < + d i < i n + l h + N + (i ; d h; n + l n) N + (i ; d h; n + l n) < If i n+l h + then Hence when i + > n + l h i + i + d n+l h i + > n + l h + i < i n + l h + we have N +(i ;dh;n+ln) N + (i ;dh;n+ln) > 3 The contruction In thi ection we will put forward a type of determinitic ening matri aociated with ubpace of F (n+l) how it i uperior to Devore contruction uing polynomial over finite field Definition 8 Given integer 0 h h h l d i < i h n+l h Let Φ 0 be the binary matri whoe row are indeed by M + (i ; d h; n+l n) whoe column are indeed by M + (i h ; d h; n + l n) with a or 0 in the (b j) poition of the matri if the b-th ubpace which belong to M + (i ; d h; n + l n) i or i not contained in the j-th ubpace which belong to M + (i h ; d h; n + l n) repectively By the Lemma 4 5 Φ 0 i an t matri whoe contant column weight i ω where (i )(n+l i ) h t (i )(n+l i ) n + l i + h h i h i + h ω (i )(i i ) h () E-ISSN: Volume 5 06

5 Theorem 9 Let 0 h h h l d i < i h n+l h Φ ω Φ 0 then Φ i a matri with coherence µ(φ) atifie i the RIP of order k with δ k (k ) whenever k < i i + Proof According to Definition 3 we know that the number of one in every column of Φ 0 i the ame that mean the value of the denominator in (4) i given which eual to ω (i )(i i ) h Hence we only need to calculate the maimum value of u b u j then we can obtain the coherence µ of Φ Let P P P t be t ditinct column of Φ For any two ditinct column P b P j thi alo mean for any two different ubpace P b P j where P b P j M + (i h ; d h; n + l n) A above-mentioned we want to know the maimum value of P b P j in other word we want to know the maimum number of ubpace U where U M + (i ; d h; n + l n) contained in both P b P j Actually ince the interection of any two ubpace i alo a ubpace hence let P P b Pj where P i the left one i not contained in E then P M + (i h ; d h; n+l n) Hence the maimum number of ubpace U which contained in P i eual to (i )(i i ) definition of coherence then µ(φ) h (i )(i i ) h (i )(i i ) h At lat a the i () By the Lemma 3 then Φ atifie the RIP of order k with δ k (k ) whenever k < i i + Thi complete the proof Eample 0 Given a contruction of ening matri over F let i i h (i h ) n + l h Then we obtain an t matri Φ with (i h )(i h +h+) h h t (i h )(i h +h) i h + µ(φ) i h h Φ atifie the RIP of order k < i h + So we can get that t (i h+ )( i h+ ) 3 (i h ) h then i h 48t t 8 Φ can be ued to recover ignal eactly with parity 3t + 6 k 3t 8 Net Let recall the Devore contruction8 Devore provide a kind of determinitic ening matri uing polynomial over finite field For our comparion we conider finite field of prime power order Let F be a finite field where i a prime power Given an integer r where 0 < r < let P r denote the et {f() (f()) r F } Then there are t : r+ uch polynomial in it Denote a null matri by H with large order the poition of H leicographically a (0 0) (0 ) ( ) ( ) We claify the contruction a three tep Firt inert one to a poition of every row of H by the following way Look Q() a a mapping of F F where Q P r F then change the value of poition ( Q()) into Every row eactly ha a one Second Tranform H into a column vector v Q with large where Note that there are eactly one in v Q ; one in the firt entrie one in the net entrie o on Third Recycle the above two tep for all the polynomial which belong to P r Hence there are t : r+ column vector At lat we obtain the matri Φ 0 with t large Lemma 8 Suppoe the matri Φ Φ 0 then Φ atifie the RIP with δ (k )r/ for any k < /r + Actually Devore polynomial determinitic matri ha been tudied by many epert They come up with that it ha ome deficiencie Firt due to the retriction of the finite field contruction approach the value range of meaurement i limited Second the time of contruction i long Third every column eactly ha nonzero entrie Hence there are more nonzero value with the greater matrice which make the parity of the matrice advere Moreover according to the practical application of image proceing we find that it reult of recontruction i uperior to Gauian matri but inferior to Hadamard matri (ee 5) Inpired by the hortcoming of Devore polynomial determinitic ening matri we need to take ening matrice into conideration generally Here we denote a ω/ which mean the rate of nonzero entrie in every column of ening matrice Given E-ISSN: Volume 5 06

6 the upper bound value of k the ening matri i better when the parameter of pare meaurement a i - maller which conduce to the recovery of ignal (ee 6) the matri i alo better when the value of t i larger which mean thi matri can provide powerfully compreed performance Conider the above-mentioned uetion carefully Then we will draw a comparion between our contruction Devore one how our better propertie than Devore one Let (i h ) n + l h h h then by Theorem 9 () we will obtain an t ening matri Φ with (i )(i i ) h t (i )i i h i + i h i + k i t h h (i )i i h i + h i h i + h h (i )(i i ) h ih + h i + ( i ) ( i ) (i )(i i ) (i )(i i ) i h i + ii h + i h i + i i h i + ii h + i h i + i ( i ) ( i ) i i h i ih + h i i + (i i )(i h i + )+(h )(h i ) a ω (i )(i i ) h (i )(i i ) h (i )i (i )i (i )i +h h ih + h ih + h i ih + h i ih + ( i ) ( i ) Note that the value of a ha been enlarged to / (i )i +h t the value of ha been reduced to (i i )(i h i + )+(h )(h i ) In order to be convenient for comparion make ure the abolutene of reult here we may well let a / (i )i +h t (i i )(i h i + )+(h )(h i ) Furthermore we alo have an t matri Φ of Devore with t r+ t r k r a where < r < i a prime power Theorem Given the matrice Φ Φ Suppoe k k are eual to their upper bound value repectively which mean k i k r Let them be eual to each other then a < a when r < (i )i +h i + Proof Since kk then i r Since a then a r Compare i a r i a / (i )i +h We have a < a when r < (i )i +h i + Theorem 3 Given the matrice Φ Φ Suppoe the value of t t are the ame then a < a when r > (i i )(i h i + ) (i )i + h + (h )(h i ) (i )i + h + Proof Since t t o r (i i )(i h i + )+(h )(h i ) E-ISSN: Volume 5 06

7 hence we get (i i )(i h i + )+(h )(h i )/(r ) Since a / we alo obtain a / (i i )(i h i + )+(h )(h i )/(r ) Compare a / (i )i +h a / (i i )(i h i + )+(h )(h i )/(r ) Hence our parameter of pare meaurement a i better when r > (i i )(i h i + )+(h )(h i ) (i )i + h + For given two ening matrice one i our contruction another i Devore contruction Let them atify the cae of Theorem 3 imultaneouly Then our contruction ha the better parity of ening matrice when (i i )(i h i + )+(h )(h i ) (i )i + h + < r < (i )i +h i + Theorem 4 Given the matrice Φ Φ Suppoe the value of t t be eual to the value of then the upper bound value of k i maller than k one when r (i i )(i h i + ) (i )i + h + (h )(h i ) (i )i + h + Proof Since t t then r (i i )(i h i + )+(h )(h i ) hence we obtain (i i )(i h i + )+(h )(h i )/(r ) For the ake of convenience here we may let (i i )(i h i + )+(h )(h i ) (r ) where > 0 then we can obtain < r (i i )(i h i + )+(h )(h i ) + < Since k < r k < + then we can alo get (i i )(i h i + )+(h )(h i ) + + Draw a comparion between (i i )(i h i + )+(h )(h i ) + + i + In other word we will compare k k upper bound value We notice that the upper bound value of k will decreae with the value of r (i i )(i h i + )+(h )(h i ) + increaing So let r (i i )(i h i + )+(h )(h i ) + < then we will obtain i > (i i )(i h i + )+(h )(h i ) + when i A above-mentioned we will obtain (i i )(i h i + )+(h )(h i ) (r ) Hence our contruction i better when i r (i i )(i h i + )+(h )(h i ) i + We have proved that our contruction i uperior to the contruction of Devore under ome condition By changing the number of parameter we will obtain a type of different determinitic ening matrice We will end up thi paper with a comparion between a matri formed by ingular linear pace over F a Devore ening matri formed by polynomial over F 3 via numerical imulation For a ignal we chooe OMP to olve l 0 -minimization () denote the olution by Define the recontruction ignal-to-noie ratio (SNR)7 of a SNR() 0 log 0 ( )db (3) If SNR () i no le than 00 db we ay the recovery of i perfect Conider a matri over F where n + l h 4 i h (i ) d i h h 0 Then we obtain an t ening matri with 8 t 8 Furthermore by Devore contruction E-ISSN: Volume 5 06

8 Figure : Perfect recovery percentage of a 8 8 matri formed by ingular linear pace over F that of a 9 7 Devore ening matri over F 3 For each k 5000 input ignal are ued to compute the percentage uing polynomial over finite field when r we alo get an t ening matri over F 3 with 9 t 7 Figure how the perfect recovery percentage of thoe two matrice For each parity k we input 5000 rom ignal to compute the perfect recovery percentage Acknowledgement Thi work i upported by the National Natural Science Foundation of China under Grant No67906 upported by the Fundamental Reearch Fund for the Central Univeritie (305L008) Reference: K Long Intercept of Freuency Agility Signal uing Coding Nyuit Folding Receiver WSEAS Tranaction on Signal Proceing 7 0 pp8-9 D Donoho Compreed ening IEEE Tran Inf Theory 5(4) 006 pp E Cé J Romberg T Tao Robut uncertainty principle: Eact ignal recontruction from highly incomplete freuency information IEEE Tran Inf Theory 5() 006 pp B K Natarajan Spare approimate olution to linear ytem SIAM J Comput 4() 995pp A Keller l p -Norm Minimization Method for Solving Nonlinear Sytem of Euation WSEAS Tranaction on Mathematic 3 04 pp X Ren J Wang S Lin A Spare Signal Recontruction Perpective for Direction-of-Arrival Etimation with Minimum Redundancy Linear Array WSEAS Tranaction on Signal Proceing 0 04 pp J Tropp A Gilbert Signal recovery from rom meaurement via orthogonal matching puruit IEEE Tran Inf Theory 53() 007 pp E Cé T Tao Near-optimal ignal recovery from rom projection: Univeral encoding - trategie IEEE Tran Inf Theory 5() 006 pp E Cé J Romberg T Tao Stable ignal recovery from incomplete inaccurate meaurement Communication on Pure Applied Mathematice 59(8) 006 pp Y Taig D Donoho Etenion of compreed ening Signal Proceing 86(3) 006 pp E Cé T Tao Decoding by linear programming IEEE Tran Inf Theory 5() 005 pp E Cé The retricted iometry property it implication for compreed ening Compte Rendu Math Acad Sci Pari 346(9-0) 008 pp M Davenport M Wakin Analyi of orthogonal matching puruit uing the retricted iometry property IEEE Tran Inf Theory 56(9) 00 pp A Cohen W Dahmen R DeVore Compreed ening bet k-term approimation J Amer Math Soc () 009 pp-3 5 J Troop Greed i good: Algorithmic reult for pare approimation IEEE Tran Inf Theory 50(0) 004 pp3-4 6 J Bourgain S Dilworth K Ford S Konyagin D Kutzarova Eplicit contruction of RIP matrice related problem Duke Math J 59() 0 pp L Welch Lower bound on the maimum cro correlation of ignal IEEE Tran Inf Theory 0(3)974 pp R Devore Determinitic contruction of compreed ening matrice J Compleity 3(4-6) 007 pp S Li F Gao G Ge S Zhang Determinitic contruction of compreed ening matrice via algebraic curve IEEE Tran Inf Theory 58(8) 0 pp E-ISSN: Volume 5 06

9 0 A Amini F Marvati Determinitic contruction of binary bipolar ternary compreed ening matrice IEEE Tran Inf Theory 57(4) 0 pp A Amini V Montazerhodjat F Marvati Matrice with mall coherence uing p-ary block code IEEE Tran Signal Proceing 60() 0 pp7-8 K Wang J Guo F Li Singular linear pace it application Finite Field Appl 7 0 pp Z Wan Geometry of claical group over finite field nd edn Science Beijing 00 4 A E Brouwer A M Cohen A Neumaier Ditance-Regular Graph Springer- Verlag Berlin Heidelberg X Li Reearch on meaurement matri baed on compreed ening Beijing Jiaotong Univerity thei of Mater Degree 00: pp5-3 6 H Wu X Zhang W Chen Meaurement matrice in compreed ening theory Journal of military communication technology 33() 0 pp D Needell J Tropp CoSaMP: Iterative ignal recovery from incomplete inaccurate ample Appl Comput Harmon Anal 6(3) 009 pp30-3 E-ISSN: Volume 5 06