11 Asymptotically Pseudo-Independent Matrices
Ilya Soloveychik and Vahid TarokhDepartment of Electrical and Computer Engineering, Duke University
Abstract —We show that the family of pseudo-randommatrices recently discovered by Soloveychik, Xiang, andTarokh in their work “Symmetric Pseudo-Random Matri-ces” exhibits asymptotic independence. More specifically,any two sequences of matrices of matching sizes fromthat construction generated using sequences of differentnon-reciprocal primitive polynomials are asymptoticallyindependent.
Index Terms —Pseudo-random matrices, asymptotic in-dependence, Wigner’s ensemble.
I. I
NTRODUCTION
Random matrices have been a very active area of re-search for the last few decades and have found enormousapplications in various areas of modern mathematics,physics, engineering, biological modeling, and otherfields [1]. In this article, we focus on the classical modelof square symmetric matrices with + − entries, referredto as square symmetric sign matrices. For this class ofmatrices, Wigner [2, 3] demonstrated that if the elementsof the upper triangular part (including the main diagonal)of an n × n matrix are independent Rademacher ( + − withequal probabilities) random variables, then as n grows aproperly scaled empirical spectral measure converges tothe semicircular law.In many engineering applications, one needs to simu-late matrices with random-looking properties. The mostnatural way to generate an instance of a random n × n sign matrix is to toss a fair coin n ( n +1)2 times, fill theupper triangular part of a matrix with the outcomesand reflect the upper triangular part into the lower.Unfortunately, for large n such an approach wouldrequire a powerful source of randomness due to theindependence condition [4]. In addition, when the datais generated by a truly random source, atypical non-random looking outcomes have non-zero probability ofshowing up. Yet another issue is that any experimentinvolving tossing a coin would be impossible to repro-duce exactly. All these reasons stimulated researchersand engineers from different areas to seek for approaches This work was supported by the Office of Naval Research grantNo. N00014-18-1-2244. of generating random-looking data usually referred toas pseudo-random sources or sequences of binary digits[5, 6]. A wide spectrum of pseudo-random numbergenerating algorithms have found applications in a largevariety of fields including radar, digital signal processing,CDMA, coding theory, cryptographic systems, MonteCarlo simulations, navigation systems, scrambling, etc.[5].The term pseudo-random is used to emphasize that thebinary data at hand is indeed generated by an entirelydeterministic causal process but its statistical proper-ties resemble some of the properties of data generatedby tossing a fair coin. Remarkably, most efforts werefocused on one dimensional pseudo-random sequences[5, 6] due to their natural applications and to the rel-ative simplicity of their analytical treatment. One ofthe most popular methods of generating pseudo-randomsequences is due to Golomb [6] and is based on linear-feedback shift registers capable of generating pseudo-random sequences (also called maximal or m -sequences)of very low algorithmic complexity [7, 8]. The studyof pseudo-random arrays and matrices was launchedaround the same time [9–12]. Among the known twodimensional pseudo-random constructions the most pop-ular are the so-called perfect maps [9, 13, 14], and twodimensional cyclic codes [11, 12]. However, except forthe recent articles [15–18], to the best of our knowledgenone of the previous works considered constructions ofsymmetric sign matrices using their spectral propertiesas the defining statistical features. In their work, theauthors of [18] designed a family of symmetric sign n × n matrices whose spectra almost surely (with respectto a certain ensemble of small size) converge to thesemicircular law when their sizes grow. The constructionis very simple and is based on binary m -sequences oflengths of the form n = 2 m − making the generationof the pseudo-random matrices very efficient and fast.The current paper is a natural extension of [18].Our goal is to show that the pseudo-random matricesconstructed in that article not only yield semicircularspectrum in the limit but also mimic the asymptoticindependence properties of the truly random Wignermatrices. We prove that if two sequences of matrices a r X i v : . [ m a t h . P R ] O c t from [18] are generated using sequences of different non-reciprocal primitive polynomials, then the former areasymptotically independent. Technically, this is achievedby verifying that the mixed centered moments of thematrices at hand vanish asymptotically. This result shedsmuch more light on the nature of spectral pseudo-randomness in matrices and provides the first example ofa family of pseudo-random matrix constructions with theaforementioned design properties and low algorithmiccomplexity.The rest of the text is organized as follows. First weset up the notation in Section II. Section III introducesthe pseudo-random construction defined in [18] and out-lines its properties relevant for the current text. SectionIV shows that our pseudo-random matrices are indeedasymptotically independent. Numerical simulations sup-porting our findings are shown in Section V. We makeour conclusions in Section VI.II. N OTATION
We denote the ranges of non-negative integers by [ n ] = { , . . . , n − } . Note also that the labeling of matrixelements starts with . We write tr ( A ) = n Tr ( A ) ,where A is an n × n matrix. Introduce a family offunctions ζ n : GF (2) n × n → {− , } n × n , { u ij } n − i,j =0 (cid:55)→ { ( − u ij } n − i,j =0 , (1)mapping binary / matrices into sign matrices of thesame sizes. Below we suppress the subscript and write ζ for simplicity. We use the following standard notationfor the limiting relations between functions. We write f ( n ) = o ( g ( n )) if lim n →∞ f ( n ) g ( n ) = 0 and f ( n ) = O ( g ( n )) if | f ( n ) | (cid:54) C | g ( n ) | for some constant C and n big enough.III. T HE P SEUDO -R ANDOM C ONSTRUCTION
In this section, we briefly outline the constructionpresented in [18].
A. Golomb Sequences
Let f ( x ) be a binary primitive polynomial of degree m and let C be a cyclic code of length n = 2 m − withthe generating polynomial h ( x ) = x n − f ( x ) . (2)In other words, C = { c ∈ GF (2) n | h ( x ) | c ( x ) } , (3) where c ( x ) = n − (cid:88) i =0 c i x i . (4)When f ( x ) is primitive, as in our case, a code con-structed in such a way is usually referred to as a simplexcode. All the non-zero codewords of the obtained codeare shifts of each other and are called Golomb sequences [6] (we can, therefore, simply say that a simplex code isgenerated by a Golomb sequence).Let C be the simplex code constructed from theprimitive binary polynomial f ( x ) as before. Fix a non-zero codeword ϕ ∈ C (a Golomb sequence) and constructa real symmetric matrix A n = { a ij } n − i,j =0 = (cid:26) √ n ( − ϕ ( i − j )+ ϕ ( j − i ) (cid:27) n − i,j =0 . (5)Matrix A n can be interpreted in the following way.Consider a circulant non-symmetric matrix T = ϕ (0) ϕ (1) ϕ (2) . . . ϕ ( n − ϕ ( n − ϕ (0) ϕ (1) . . . ϕ ( n − ϕ ( n − ϕ ( n − ϕ (0) . . . ϕ ( n − ... ... ... . . . ... ϕ (1) ϕ (2) ϕ (3) . . . ϕ (0) . (6)The consecutive rows of T are simply cyclic shifts of theGolomb sequence written in its first rows. The symmetricmatrix A n can now be written as A n = 12 √ n ζ ( T + T (cid:62) ) . (7)It is easy to check that the obtained matrix is circulant,since for any k ∈ [ n ] , A n is invariant under the shift ofindices of the form i → i + k mod n, j → j + k mod n. (8)Recall that any non-zero codeword of C is a cyclicshift of ϕ , therefore, we may obtain an ensemble ofmatrices from the code C indexed by integers within therange a ∈ [ n ] , as A n ( a ) = (cid:26) √ n ( − ϕ ( i − j + a )+ ϕ ( j − i + a ) (cid:27) n − i,j =0 , (9)with the original matrix A n corresponding to A n (0) . Definition 1.
Given a primitive binary polynomial f ( x ) ,an ensemble of pseudo-random matrices A n of order n is the set of all A n ( a ) , a ∈ [ n ] and their negatives,endowed with the uniform probability measure. Below, whenever expectation over A n is consideredit should be always treated with respect to the uniformmeasure over A n . Definition 2 ([19]) . We say that a sequence { A n } ∞ n =1 ofmatrices of growing sizes has an asymptotic eigenvaluedistribution if β r = lim n →∞ tr ( A rn ) (10) exist for all r ∈ N . One of the central results of [18] reads as follows.
Proposition 1 (Proposition 1 from [18]) . Let A n ∈ A n ,then for a fixed r ∈ N and n = n ( m ) tending to infinity, E [ β r ( A n )] = (cid:40) β r + O (cid:0) n (cid:1) , r even , , r odd , (11) where β r = (cid:90) x r dF sc = (cid:40) , r odd , C r/ r , r even , (12) are the moments of the semicircular distribution [2] and C r = (2 r )! r !( r + 1)! (13) are the Catalan numbers. This result in particular implies that the limitingspectral law of our pseudo-random matrices is Wigner’ssemicircular law.Consider a pair of sequences of matrices { A n } ∞ n =1 and { B n } ∞ n =1 , each of which is assumed to have anasymptotic eigenvalue distribution. Ideally, we wantto understand the limiting behavior of any reasonablyregular function of A n and B n . By the method ofmoment this calls for investigation of the moments tr (cid:0) A t n B s n · · · A t k n B s k n (cid:1) for natural powers t i and s i .Since our pseudo-random construction yields circulantmatrices, they commute and we only need to study theirmixed moments of the form tr (cid:0) A tn B sn (cid:1) . Definition 3.
Let { A n } ∞ n =1 and { B n } ∞ n =1 be two se-quences of random matrices of growing and matchingsizes having asymptotic eigenvalue distributions with themoments δ r and ζ r respectively. Let t, s ∈ N , we say that A n and B n are asymptotically independent if tr (cid:0)(cid:0) A tn − δ t I (cid:1) ( B sn − ζ s I ) (cid:1) → , n → ∞ . (14)Note that the mode of asymptotic independence (e.g.,in expectation, in probability, almost surely) is deter-mined by the mode of convergence to zero in (14). IV. A SYMPTOTIC P SEUDO -I NDEPENDENCE
In this section we show that two sequences of pseudo-random matrices constructed as described in Section IIIfrom different non-reciprocal primitive polynomials areasymptotically independent in expectation, namely thatthey satisfy the moment condition (14) on average overthe ensembles A n and B n .Given a binary polynomial f ( x ) , its reciprocal is apolynomial of the same degree defined as ˆ f ( x ) = x deg f f (cid:0) x − (cid:1) . (15) Lemma 1.
A reciprocal of a primitive polynomial isprimitive.Proof.
The result follows directly from the properties ofthe primitive polynomials and the fact that if ε is a rootof a polynomial, ε − is the root of its reciprocal.Assume that the generating polynomial g ( x ) of theGolomb sequence ψ does not coincide neither with thegenerating polynomial f ( x ) of φ nor with its reciprocal ˆ f ( x ) . Proposition 2.
Let { f m ( x ) } ∞ m =1 and { g m ( x ) } ∞ m =1 betwo sequences of different and non-reciprocal primitivepolynomials of degrees m . For n = 2 m − , let { A n } ∞ n =1 and { B n } ∞ n =1 be pseudo-random matrices constructedfrom f m and g m correspondingly with arbitrary seeds,then A n and B n are asymptotically independent onaverage.Proof. Our goal is to show that the expressions of theform E (cid:2) tr (cid:0)(cid:0) A tn ( a ) − β t I (cid:1) ( B sn ( b ) − β s I ) (cid:1)(cid:3) , (16)for all natural t and s converge to zero when n increases.Introduce the following quantity, E = E (cid:2) tr (cid:0)(cid:0) A tn ( a ) − β t I (cid:1) ( B sn ( b ) − β s I ) (cid:1)(cid:3) + β t β s = E (cid:2) tr (cid:0) A tn ( a ) B sn ( b ) (cid:1)(cid:3) (17) = 1 n E n − (cid:88) i,j =0 (cid:2) A t (cid:3) ij [ B s ] ij = 1 n n − (cid:88) a =0 n − (cid:88) b =0 r n r n − (cid:88) i ,...,i t − =0 n − (cid:88) j ,...,j s − =0 ,j = i t − ,j s − = i ( − γ i , j ( a,b ) , where r = t s , (18) and we denote γ i , j ( a, b ) = t − (cid:88) q =0 ϕ ( i q +1 − i q + a ) + ϕ ( i q − i q +1 + a )+ s − (cid:88) q =0 ψ ( j q +1 − j q + b ) + ψ ( j q − j q +1 + b ) , (19)where we treat the indices q of the vertices i q and j q modulo t and s , respectively. Instead of treating the ex-pression in (16), it is more convenient to demonstrate that E converges to β r β s which is equivalent to the originalstatement. Let us also write explicitly the condition onindices appearing in (17) as j = i t − , j s − = i . (20)Set u q = i q +1 − i q mod n, q = 0 , . . . , t − , (21) w q = j q +1 − j q mod n, q = 0 , . . . , s − . (22)Denote the obtained t - and s -tuples by u = ( u , . . . , u t − ) ∈ [ n ] t , (23) w = ( w , . . . , w s − ) ∈ [ n ] s , (24)and following [18] use the function ν t : [ n ] t → GF (2) n , ( u , . . . , u t − ) (25) (cid:55)→ (cid:26) t − (cid:88) q =0 ( u q = i ) + ( − u q = i ) mod 2 (cid:27) n − i =0 , where is an indicator function and the equalitiesare modulo n . We refer the reader to [18] for adetailed discussion on the properties of ν t . Briefly, ν t ( · ) takes the t -tuple u = ( u , . . . , u t − ) andfirst maps it into an extended t -tuple ( u , − u ) =( u , . . . , u t − , − u , . . . , − u t − ) ∈ [ n ] t . Then it calcu-lates the number of appearances of every number u ∈ [ n ] in this t -tuple, which we denote by { u } and constructsa codeword c ∈ GF (2) n by setting its elements withindices u to { u } mod 2 and zeros otherwise. Forconvenience, we suppress the subscript of ν t below.Rewrite γ i , j ( a, b ) as γ i , j ( a, b ) = τ ( ν ( u ) , a ) + τ ( ν ( w ) , b ) , (26)where τ ( ν ( u ); a ) (27) = (cid:80) t − q =0 (cid:2) ϕ ( u q + a ) + ϕ ( − u q + a ) (cid:3) mod 2 ,ν ( u ) (cid:54) = , ν ( u ) = . With this notation, we obtain E = 12 r n r +3 n − (cid:88) a =0 n − (cid:88) b =0 (cid:88) u , w ( − τ ( ν ( u ); a )+ τ ( ν ( w ); b ) , (28)where we assume u and w to satisfy (20). Let us denote k = j − i , (29)then (28) can be rewritten as E = 12 r n r +3 n − (cid:88) a =0 n − (cid:88) b =0 n − (cid:88) k =0 n − (cid:88) i =0 (cid:88) u k , w k ( − τ ( ν ( u ); a )+ τ ( ν ( w ); b ) , (30)where t - and s -tuples u k and w k have their elements i t − − i = k and j − j s − = k , respectively. Clearlyfor fixed k and i , the averages over a and b decoupleand we can switch the order of summation to obtain E = n − (cid:88) k =0 n n − (cid:88) i =0 (cid:34) t n t/ n − (cid:88) a =0 (cid:88) u k ( − τ ( ν ( u k ); a ) (cid:35) × (cid:34) s n s/ n − (cid:88) b =0 (cid:88) w k ( − τ ( ν ( w k ); b ) (cid:35) . (31)Now we deal with the sums in the square bracketsseparately. We focus on the first sum, the second istreated analogously. Let us consider the case of k = 0 and even t . Here, similarly to [18] we need to count thenumber of even paths starting and ending at i in orderto calculate the leading term of the expected value. Thecalculation follows the same reasoning as in [18] and forevery fixed i yields t n t/ n − (cid:88) a =0 (cid:88) u k ( − τ ( ν ( u k ); a ) = β t + O (cid:18) n (cid:19) . (32)For all other combinations of k > or odd t , using thesame approach as in the derivation of a bound on III inthe proof of Proposition 1 in [18], we get t n t/ n − (cid:88) a =0 (cid:88) u k ( − τ ( ν ( u k ); a ) = O (cid:18) n (cid:19) . (33)Similarly, for the second sum, s n s/ n − (cid:88) b =0 (cid:88) w k ( − τ ( ν ( w k ); a ) = β s + O (cid:18) n (cid:19) , (34)when k = 0 and s is even. Otherwise, s n s/ n − (cid:88) b =0 (cid:88) w k ( − τ ( ν ( w k ); a ) = O (cid:18) n (cid:19) . (35) Overall, we conclude E = n − (cid:88) k =0 n n − (cid:88) i =0 (cid:20) β t + O (cid:18) n (cid:19)(cid:21) (cid:20) β s + O (cid:18) n (cid:19)(cid:21) = β t β s + O (cid:18) n (cid:19) , (36)which according to (17) completes the proof.It is important to note that Proposition 2 claimsasymptotic independence of the two sequences at handon average. In fact, asymptotic almost sure independencecan also be demonstrated using the same technique as in[18] (see Figure 1 showing the decay of the variance).However, to avoid duplication of the proof we decide toomit the rigorous derivation here.V. N UMERICAL E XPERIMENTS
In this section, we illustrate our theoretical resultsfrom Section IV using numerical simulations. Morespecifically, we examine the behavior of low mixedmoments of our pseudo-random matrices when the sizesof the latter grow.Let us fix a range M = m b , . . . , m e of integers andconsider two sequences of primitive binary polynomials f m i and g m i , m i ∈ M . Each of the constructedpolynomials gives raise to an ensemble of cardinality n i = 2 m i − of pseudo-random matrices of sizes n i × n i .Denote the corresponding ensembles by A n i and B n i . Inour experiment we took m b = 7 , m e = 19 . Polynomials f m i were chosen to be the first polynomials in thecorresponding rows of the table [20]. Polynomials g m i were obtained through -fold decimation of f m i -s andcan be easily checked to be non-reciprocal with f m i -s[21].We focus on studying the behavior of the expectedodd mixed moment µ ts ( n i ) = E A ni ∼A ni , B ni ∼B ni tr (cid:0) A tn i B sn i (cid:1) , (37)as a function of n i . Figure 1 demonstrates that the mixedmoments at hand decay to zero as expected. In addition,it shows the decay of the variance of the trace in (37),which implies almost sure asymptotic independence asexplained earlier.Figure 2 provides an empirical comparison of the ratesof convergence of higher mixed moments to zero. Here,we took two polynomials f jm i and g jm i , j = 1 , of everydegree in the range defined by m b = 7 , m e = 17 fromthe same table [20] and averaged the moments over thetwo corresponding ensembles A jn i and B jn i . Remarkably,this graph supports our theoretical result established inProposition 2 claiming that mixed moments decay withthe rate of O (cid:0) n (cid:1) . n -6 -4 -2 First mixed moment +std Fig. 1. First mixed moment of the form (37) plus its standarddeviation region in pseudo-random matrices of sizes n = 2 − , . . . , − . n -6 -4 Mixed moments
Fig. 2. Higher mixed moments of the form (37) in pseudo-randommatrices of sizes n = 2 − , . . . , − . VI. C
ONCLUSIONS
In this article, we show that the recently discovered in[18] family of pseudo-random symmetric sign matricesexhibits asymptotic independence properties. This resultsallows one to generate pairs of random-looking symmet-ric sign matrices with semicircular limiting spectrum andvanishing odd mixed moments.R
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