The dominant eigenvalue of an essentially nonnegative tensor

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1 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Lnear Algebra Appl. 2013; 00:1 13 Publshed onlne n Wley InterScence ( The domnant egenvalue of an essentally nonnegatve tensor L. P. Zhang a L. Q. Q b Z. Y. Luo c Y. Xu b a Department of Mathematcal Scences, Tsnghua Unversty, Bejng, Chna b Department of Appled Mathematcs, The Hong Kong Polytechnc Unversty, Hong Kong c State Key Laboratory of Ral Traffc Control and Safety, Bejng Jaotong Unversty, Bejng, Chna SUMMARY It s well known that the domnant egenvalue of a real essentally nonnegatve matrx s a convex functon of ts dagonal entres. Ths convexty s of practcal mportance n populaton bology, graph theory, demography, analytc herarchy process and so on. In ths paper, the concept of essentally nonnegatvty s extended from matrces to hgher order tensors, and the convexty and log convexty of domnant egenvalues for such a class of tensors are establshed. Partcularly, for any nonnegatve tensor, the spectral radus turns out to be the domnant egenvalue and hence possesses these convextes. Fnally, an algorthm s gven to calculate the domnant egenvalue, and numercal results are reported to show the effectveness of the proposed algorthm. Copyrght c 2013 John Wley & Sons, Ltd. Receved... KEY WORDS: essentally nonnegatve tensor; domnant egenvalue; convex functon; spectral radus; algorthm 1. INTRODUCTION Tensors are ncreasngly ubqutous n varous areas of appled, computatonal, and ndustral mathematcs and have wde applcatons n data analyss and mnng, nformaton scence, sgnal/mage processng, and computatonal bology, etc; see the workshop report [1] and references theren. A tensor can be regarded as a hgher-order generalzaton of a matrx, whch takes the form A = (A 1 m ), A 1 m R, 1 1,..., m n. Such a mult-array A s sad to be an m-order n-dmensonal square real tensor wth n m entres A 1 m. In ths regard, a vector s a frst-order tensor and a matrx s a second-order tensor. Tensors of order more than two are called hgher-order tensors. Analogous wth that of matrces, the theory of egenvalues and egenvectors s one of the fundamental and essental components n tensor analyss. 72 references on egenvalues of tensors can be found n the bblography [2]. Wde range of practcal applcatons can be found the references there. Compared wth that of matrces, egenvalue problems for hgher-order tensors are nonlnear due to ther multlnear structure. Varous types of egenvalues are defned for hgher-order tensors n the settng of multlnear algebra. For example, the egenvalue, the H-egenvalue, the E- egenvalue, the Z-egenvalue, the N-egenvalue defned by Q for even order symmetrc tensors [3], the l p egenvalues for general order symmetrc tensors, and the mode- egenvalues for general square tensors defned by Lm [4], the M-egenvalue for a partally symmetrc fourth-order tensor, Correspondence to: lzhang@math.tsnghua.edu.cn (L. P. Zhang); maqlq@polyu.edu.hk (L. Q. Q); starkeynature@hotmal.com (Z. Y. Luo); y.xu1983@gmal.com (Y. Xu) Copyrght c 2013 John Wley & Sons, Ltd. Prepared usng nlaauth.cls [Verson: 2010/05/13 v2.00]

2 2 L. P. ZHANG defned by Q, Da and Han [5], the D-egenvalue for a fourth-order symmetrc tensor and a secondorder symmetrc tensor, defned by Q, Wang and Wu [6], egenvalues of general square tensors extended by Q n [2] Chang, Pearson and Zhang n [8] and equvalent egenvalue par classes by Cartwrght and Sturmfels [7]. Here, we are concerned wth the one n [2, 8] as revewed below. Defnton 1.1 Let C be the complex feld. For a vector x C n, we use x to denote ts components, and x [m 1] to denote a vector n C n such that x [m 1] = x m 1 for all. Ax m 1 denotes a vector n C n, whose th component s 2,..., m =1 A 2 m x 2 x m. A par (λ, x) C (C n \{0}) s called an egenvalue-egenvector par of A, f they satsfy: Ax m 1 = λx [m 1]. (1) Nonnegatve tensors, arsng from multlnear pagerank [4], spectral hypergraph theory [9, 10, 11], and hgher-order Markov chans [12], etc., form a sngularly mportant class of tensors and have attracted more and more attenton snce they share some ntrnsc propertes wth those of the nonnegatve matrces. One of those propertes s the Perron-Frobenus theorem on egenvalues. In [13], Chang, Pearson, and Zhang generalzed the Perron-Frobenus theorem for nonnegatve matrces to rreducble nonnegatve tensors. In [14], Fredland, Gaubert and Han generalzed the Perron-Frobenus theorem to weakly rreducble nonnegatve tensors. Further generalzaton of the Perron-Frobenus theorem to nonnegatve tensors can be found n [15]. Numercal methods for fndng the spectral radus of nonnegatve tensors are subsequently proposed. Ng, Q, and Zhou [12] provded an teratve method to fnd the largest egenvalue of an rreducble nonnegatve tensor by extendng the Collatz method [16] for calculatng the spectral radus of an rreducble nonnegatve matrx. The Ng-Q-Zhou method s effcent but t s not always convergent for rreducble nonnegatve tensors. Chang, Pearson and Zhang [17] extended the noton of prmtve matrces nto the realm of tensors, and establshed the convergence of the Ng-Q-Zhou method for prmtve tensors. Zhang and Q [18] establshed global lnear convergence of the Ng-Q-Zhou method for essentally postve tensors. Lu, Zhou and Ibrahm [19] proposed an always convergent algorthm for computng the largest egenvalue of an rreducble nonnegatve tensors. Zhang, Q, and Xu [20] establshed ts explct lnear convergence rate for weakly postve tensors. The essentally nonnegatve tensor we defned n ths paper s ultmately related to the nonnegatve tensor and ncludes the latter one as a specal case. It s a hgher order generalzaton of the socalled essentally nonnegatve matrx, whose off-dagonal entres are all nonnegatve. Such a class of matrces possesses nce propertes on egenvalues. It follows from the famous Perron-Frobenus theorem for nonnegatve matrces that for any essentally nonnegatve matrx A, there exsts a real egenvalue wth a nonnegatve egenvector, whch s the largest one among real parts of all other egenvalues of A. Ths specal egenvalue, termed as r(a), s often called the domnant egenvalue of A. Moreover, r(a) s known as a convex functon of the dagonal entres of A. Ths convexty s a fundamental property for essentally nonnegatve matrces [21, 22, 23] and has numerous applcatons, not only n many branches of mathematcs, such as graph theory [24], dfferental equatons [23], but also n practcal felds, e.g., populaton bology [23] and analytc herarchy process [25] as well. A natural queston arses: does ths convexty mantan for hgher-order essentally nonnegatve tensors? In ths paper, we wll gve an affrmatve answer to ths queston. Smlar to the essentally nonnegatve matrx, an essentally nonnegatve tensor has a real egenvalue wth the property that t s greater than or equal to the real part of every egenvalue of A. We also call t the domnant egenvalue of A, and denoted by λ(a). Partcularly, f A s nonnegatve, we have ρ(a) = λ(a), where ρ(a) s the spectral radus of A. By employng the technque proposed n [23], we manage to obtan that the domnant egenvalue s a convex functon of the dagonal Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

3 THE DOMINANT EIGENVALUE OF AN ESSENTIALLY NONNEGATIVE TENSOR 3 elements for any essentally nonnegatve tensor. In addton, t s also a convex functon of all elements of a tensor n some specal convex set of tensors. Furthermore, the log convexty s also exploted for essentally nonnegatve tensors wth whose entres are ether dentcally zero or log convex of some real unvarate functons. Fnally, we propose an algorthm to calculate the domnant egenvalue, convergence of the proposed algorthm s establshed and numercal results are reported to show the effectveness of the proposed algorthm. Ths paper s organzed as follows. In Secton 2, we recall some prelmnary results, ntroduce the concept of essentally nonnegatve tensors, and characterze some basc propertes of such tensors. In Secton 3, we show that the spectral radus of nonnegatve tensors s a convex functon of the dagonal elements, and so s the domnant egenvalue of essentally nonnegatve tensors. Secton 4 s devoted to the log convexty of the domnant egenvalue. In Secton 5, we gve an algorthm to calculate the domnant egenvalue, and some numercal results are reported. An applcaton and some concludng remarks are made n Secton PRELIMINARIES AND ESSENTIALLY NONNEGATIVE TENSORS We start ths secton wth some fundamental notons and propertes on tensors. An m-order n-dmensonal tensor A s called nonnegatve (or, respectvely, postve) f A 1 m 0 (or, respectvely, A 1 m > 0). The m-order n-dmensonal unt tensor, denoted by I, s the tensor whose entres are δ 1... m wth δ 1... m = 1 f and only f 1 = = m and otherwse zero. The symbol A B means that A B s a nonnegatve tensor. A tensor A s called reducble, f there exsts a nonempty proper ndex subset I {1, 2,..., n} such that A 1 m = 0, 1 I, 2,..., m I. Otherwse, we say A s rreducble. We call ρ(a) the spectral radus of tensor A f ρ(a) = max{ λ : λ s an egenvalue of A}, where λ denotes the modulus of λ. An mmedate consequence on the spectral radus follows drectly from Corollary 3 n [3]. Lemma 2.1 Let A be an m-order n-dmensonal tensor. Suppose that B = a(a + bi), where a and b are two real numbers. Then µ s an egenvalue of B f and only f µ = a(λ + b) and λ s an egenvalue of A. In ths case, they have the same egenvectors. Moreover, ρ(b) a (ρ(a) + b ). Let P := {x R n : x 0, 1 n}, and nt(p ) = {x R n : x > 0, 1 n}. The Perron-Frobenus theorem for nonnegatve tensors s as below, followng by [13, Theorem 1.4]. Theorem 2.1 If A s an rreducble nonnegatve tensor of order m and dmenson n, then there exst λ 0 > 0 and x 0 nt(p ) such that Ax0 m 1 = λ 0 x [m 1] 0. Moreover, f λ s an egenvalue wth a nonnegatve egenvector, then λ = λ 0. If λ s an egenvalue of A, then λ λ 0. The well-known Collatz mnmax theorem [16] for rreducble nonnegatve matrces has been extended to rreducble nonnegatve tensors n [13, Theorem 4.2]. Theorem 2.2 Assume that A s an rreducble nonnegatve tensor of order m dmenson n. Then mn max (Ax m 1 ) x nt(p ) x >0 x m 1 = λ 0 = max mn x nt(p ) x >0 (Ax m 1 ) x m 1, where λ 0 s the unque postve egenvalue correspondng to a postve egenvector. Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

4 4 L. P. ZHANG For nonnegatve tensors, Yang and Yang [15] asserted that the spectral radus s an egenvalue, whch s a generalzaton of the weak Perron-Frobenus theorem for nonnegatve matrces. We state t [15, Theorem 2.3 and Lemma 5.8] n the followng theorem. Theorem 2.3 Assume that A s a nonnegatve tensor of order m dmenson n, then ρ(a) s an egenvalue of A wth a nonzero nonnegatve egenvector. Moreover, for any x nt(p ) we have (Ax m 1 ) mn 1 n x m 1 ρ(a) max 1 n (Ax m 1 ) x m 1. The followng nequalty and contnuty of the spectral radus were gven n [15, Lemma 3.5] and the proof of [15, Theorem 2.3], respectvely. Lemma 2.2 Let A be a nonnegatve tensor of order m and dmenson n, and ε > 0 be a suffcently small number. Suppose A B, then ρ(a) ρ(b). Furthermore, f A ε = A + E where E denotes the tensor wth every entry beng ε, then lm ε 0 ρ(a ε) = ρ(a). Based on the above results, we can easly get the followng lemma. Lemma 2.3 Suppose that A s an rreducble nonnegatve tensor of order m dmenson n and that there exsts a nonzero vector x P and a real number β such that Ax m 1 βx [m 1]. (2) Then β > 0, x nt(p ), and ρ(a) β. Furthermore, ρ(a) < β unless equalty holds n (2). Assume on the contrary that for x nt(p ) there exsts a nonempty proper ndex subset I {1, 2,..., n} such that x = 0 for I and x > 0 for I. It follows from (2) that A 1 m = 0, 1 I, 2,..., m I. A contradcton to the rreducblty of A comes, whch henceforth mples that x nt(p ). Together wth Lemma 2.2 n [12], Ax m 1 nt(p ) s establshed. It further deduces that β > 0, and then the last statement holds from Lemma 5.9 n [15]. Ths completes the proof. A smple but useful result follows mmedately from Lemmas 2.2 and 2.3. Lemma 2.4 Let A and B be rreducble nonnegatve tensors of order m dmenson n. If A B and A = B, then ρ(a) < ρ(b). By Lemma 2.2, ρ(a) ρ(b). Snce B s rreducble, Theorem 2.1 mples that there exsts x nt(p ) such that Ax m 1 Bx m 1 = ρ(b)x [m 1]. (3) Snce x nt(p ) and A = B, equalty cannot hold n (3). The desred strct nequalty ρ(a) < ρ(b) holds from Lemma 2.3. The remanng of ths secton s devoted to the essentally nonnegatve tensor, wth the ntroducton of ts defnton and some basc propertes. Defnton 2.1 Let A be an m-order and n-dmensonal tensor. A s sad to be essentally nonnegatve f all ts off-dagonal entres are nonnegatve. Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

5 THE DOMINANT EIGENVALUE OF AN ESSENTIALLY NONNEGATIVE TENSOR 5 Theorem 2.4 Let A be an m-order and n-dmensonal essentally nonnegatve tensor. Then there exsts α > 0 such that αi + A s nonnegatve. Moreover, A has a real egenvalue λ(a) wth correspondng egenvector n P and λ(a) Reλ for every egenvalue λ of A. Furthermore, Take λ(a) = ρ(αi + A) α. α = max 1 n A Clearly, α > 0 and αi + A s nonnegatve. By Lemma 2.1 and Theorem 2.3, we have ρ(αi + A) = α + λ 1, (4) where λ 1 s an egenvalue of A wth correspondng egenvector n P. Thus, (4) mples λ 1 R. Let λ(a) = λ 1, It follows from Lemma 2.1 that, The desred result arrves. λ(a) + α = max{ α + λ : λ s an egenvalue of A} α + λ α + Reλ. We call such an egenvalue n the above theorem the domnant egenvalue of A. Throughout ths paper, ρ(a) and λ(a) wll denote the spectral radus and domnant egenvalue respectvely of a tensor A. In the next secton, we wll show that both ρ(a) and λ(a) are convex functons of the dagonal elements of A. 3. CONVEXITY OF THE SPECTRAL RADIUS AND THE DOMINANT EIGENVALUE Based on Theorems 2.1 and 2.3, we proceed wth the convexty of the domnant egenvalue of essentally nonnegatve tensors n ths secton. It can be verfed that the dagonal entres have nothng to do wth the rreducblty of a tensor. Specfcally, let A be an essentally nonnegatve tensor of order m and dmenson n, defne a nonnegatve tensor B by B 1... m = 0 f 1 = = m and the others are A 1... m. Then A s rreducble f and only f B s. Equvalently, A s rreducble f and only f A + αi s, whenever t s nonnegatve. Thus, by Lemma 2.2 and Theorem 2.4, t s suffcent to consder the class of rreducble nonnegatve tensors. Theorem 3.1 If A s a gven rreducble nonnegatve tensor of order m and dmenson n, and D s allowed to vary n the class of nonnegatve dagonal tensors, then the spectral radus ρ(a + D) s a convex functon of the dagonal entres of D. That s, for nonnegatve dagonal tensors C and D we have ρ(a + tc + (1 t)d) tρ(a + C) + (1 t)ρ(a + D), t [0, 1]. (5) Moreover, equalty holds n (5) for some t (0, 1) f and only f D C s a scalar multple of the unt tensor I. Snce both A + C and A + D are rreducble nonnegatve tensors, by Theorem 2.1 and Theorem 2.3 we have ρ(a + C) > 0, ρ(a + D) > 0, and there exst x, y nt(p ) such that (A + C)x m 1 = ρ(a + C)x [m 1], (A + D)y m 1 = ρ(a + D)y [m 1]. Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

6 6 L. P. ZHANG That s, for = 1, 2,..., n we have ρ(a + C) = C... + ρ(a + D) = D m = m=1 A 2... m x 2 x m x, A 2... m y 2 y m y, and hence ρ(a + C) C... > 0 and ρ(a + D) D... > 0. The nequalty between geometrc and arthmetc means yelds ( ) t ( 2... m =1 A 2... m x 2 x m x 2... m =1 A 2... m y 2 y m y Therefore, Hölder s nequalty and Theorem 2.2 gve from (6) { ρ(a + tc + (1 t)d) max 1 n tc... + (1 t)d... + tρ(a + C) + (1 t)ρ(a + D), ) 1 t t(ρ(a + C) C...) +(1 t)(ρ(a + D) D... ). (6) 2... m =1 A 2... m z 2 z m z where z = x t y1 t for = 1,..., n. Ths shows (5) holds. The nequalty between geometrc and arthmetc means mples that equalty n (5) holds for t (0, 1) f and only f ρ(a + C) C... = ρ(a + D) D... for = 1,..., n,.e., D C = γi where γ = ρ(a + D) ρ(a + C). Ths completes the proof. The convexty nvolved n Theorem 3.1 can be extended to the case of essentally nonnegatve tensors as follows. Corollary 3.1 If A s a gven rreducble essentally nonnegatve tensor of order m dmenson n and D s allowed to vary n the class of dagonal tensors, then the domnant egenvalue λ(a + D) s a convex functon of the dagonal entres of D. That s, for dagonal tensors C and D we have λ(a + tc + (1 t)d) tλ(a + C) + (1 t)λ(a + D), t [0, 1]. (7) Moreover, equalty holds n (7) for some t (0, 1) f and only f D C s a scalar multple of the unt tensor I. Take α = 1 + max 1 n { A... + C... + D... }. Then αi + A + C and αi + A + D are all rreducble nonnegatve tensors. By Theorem 2.4 and Theorem 3.1, we have for 0 t 1 λ(a + tc + (1 t)d) + α = ρ(αi + A + tc + (1 t)d) whch yelds (7). Ths completes the proof. tρ(αi + A + C) + (1 t)ρ(αi + A + D) = tλ(a + C) + (1 t)λ(a + D) + α, Invokng the contnuty presented n Lemma 2.2, t s easy to see that Theorem 3.1 and Corollary 3.1 hold even when A s reducble. Moreover, Theorem 3.1 and Corollary 3.1 gve necessary and suffcent condtons for the strct convexty. It s worth pontng out that the convexty of } Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

7 THE DOMINANT EIGENVALUE OF AN ESSENTIALLY NONNEGATIVE TENSOR 7 the domnant egenvalue only works on the dagonal elements rather than on all elements of the essentally nonnegatve tensor, except for some specal cases. By collectng all symmetrc essentally nonnegatve tensors of order m and dmenson n, we can get a closed convex cone, says S(m, n). The domnant egenvalue of any tensor n S(m, n) remans convex of all elements of the correspondng tensor n the doman S(m, n), as the followng proposton shows. Proposton 3.1 For any A, B S(m, n), and any t [0, 1], we have λ(ta + (1 t)b) tλ(a) + (1 t)λ(b). For any A, B S(m, n), there exsts an nteger k > 0 such that A + ki and B + ki are nonnegatve and symmetrc and hence for any of ther convex combnatons. The Perron-Frobenus theorem then ensures that ρ(a + ki), ρ(b + ki) and ρ(ta + (1 t)b + ki) (t [0, 1]) all act as egenvalues of the correspondng nonnegatve symmetrc tensor. By the varatonal approach, t follows that ρ(ta + (1 t)b + ki) { = max t max (ta + (1 t)b + ki)x m : { (A + ki)x m : } x m = 1 =1 } { x m = 1 + (1 t) max (B + ki)x m : =1 = tρ(a + ki) + (1 t)ρ(b + ki). Combnng wth the fact that ρ(a + ki) = λ(a) + k, the desred convexty follows. } x m = 1 =1 4. LOG CONVEXITY OF THE SPECTRAL RADIUS AND THE DOMINANT EIGENVALUE If a functon f(x) s postve on ts doman and log f(x) s convex, then f(x) s called log convex. It s known that the sum or product of log convex functons s also log convex. In ths secton we extend Kngman s theorem [23] for matrces to tensors. Our motvaton for the followng proof comes from [23]. Theorem 4.1 For t [0, 1] assume that F(t) = (F 1... m (t)) s an m-order n-dmensonal rreducble nonnegatve tensor, and suppose that for 1 1,..., m n, F 1... m (t) s ether dentcally zero or postve and a log convex functon of t. Then ρ(f(t)) s a log convex functon of t for t [0, 1]. That s, f F(0) = A, F(1) = B, and a nonnegatve tensor G(t) = ( ) A 1 t 1... m B t 1... m, then ρ(f(t)) ρ(g(t)) ρ(a) 1 t ρ(b) t. (8) Moreover, the frst equalty occurs n (8) for some t wth t (0, 1) f and only f F(t) = G(t), and the second equalty occurs n (8) for some t wth t (0, 1) f and only f there exsts a constant σ > 0 and a postve dagonal matrx D = dag(d 1,..., d n ) such that m 1 {}}{ B = σa D (m 1) D D wth B m = σa m d (m 1) 1 d 2 d m. Clearly, G(0) = F(0) = A and G(1) = F(1) = B. The log convexty assumpton on F 1... m (t) Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

8 8 L. P. ZHANG mples that, for t [0, 1], F(t) G(t), whch, together wth Lemma 2.2, mples ρ(f(t)) ρ(g(t)). (9) Snce F(t) s rreducble, f equalty holds n (9) for some t 0 wth 0 < t 0 < 1, Lemma 2.4 mples that F(t 0 ) = G(t 0 ). Snce F(0) and F(1) are rreducble nonnegatve, Theorem 2.1 shows that there exst x, y nt(p ) such that Ax m 1 = ρ(a)x [m 1], By m 1 = ρ(b)y [m 1]. For a fxed t (0, 1), defne z = x 1 t y t,.e., z = x 1 t of G(t)z m 1 satsfes y t for 1 n. Then, the th component Hence, Hölder s nequalty gves ( G(t)z m 1 ) ( G(t)z m 1 ) = ( 2... m=1 It follows from Lemma 2.3 and (10) that 2... m =1 A 1 t 2... m B t 2... m z 2 z m. A 2... m x 2 x m ) 1 t ( 2... m=1 B 2... m y 2 y m ) t = ρ(a) 1 t ρ(b) t z m 1. (10) ρ(g(t)) ρ(a) 1 t ρ(b) t. Furthermore, equalty holds n (10) for some t (0, 1) f and only f, for 1 n, Summng (11) over 2... m yelds B 2... m y 2 y m = σ A 2... m x 2 x m. (11) ρ(b)y m 1 = σ ρ(a)x m 1. (12) Take Then, combnng (11) and (12) we obtan σ = ρ(b) ρ(a), d = x y, B 2... m = σa 2... m d (m 1) d 2 d m,.e., m 1 {}}{ B = σa D (m 1) D D. Ths completes the proof. By Theorems 2.3 and 2.4, the above theorem also holds for the domnant egenvalue of F(t), when F(t) s essentally nonnegatve wth t [0, 1]. Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

9 THE DOMINANT EIGENVALUE OF AN ESSENTIALLY NONNEGATIVE TENSOR 9 5. AN ALGORITHM FOR CALCULATING THE DOMINANT EIGENVALUE Let A be an essentally nonnegatve tensor of order m and dmenson n. In ths secton we propose an algorthm to calculate the domnant egenvalue of an essentally nonnegatve tensor. Ths algorthm s a modfcaton of the Ng-Q-Zhou algorthm gven n [12]. By Lemma 2.2 and Theorem 2.4, we modfy the Ng-Q-Zhou algorthm such that for any essentally nonnegatve tensor, the sequence generated by the modfed algorthm always converges to ts domnant egenvalue. Defne two functons from nt(p ) to P : F (x) := mn x 0 (Wx m 1 ) x m 1, G(x) := max x 0 (Wx m 1 ) x m 1, (13) where W s an rreducble nonnegatve tensor. The detals of the modfed algorthm are gven as follows. Algorthm 5.1: Step 0. Gven a suffcently small number ε > 0. Let where W = A + αi + E, (14) α = max 1 n A , and E s the tensor wth every entry beng ε. Choose any x (0) nt(p ). Set y (0) = W ( x (0)) m 1 and k := 0. Step 1. Compute x (k+1) = ( ) 1 y (k) [ m 1 ] ( y (k)) 1 [ ], m 1 ( ) m 1 y (k+1) = B x (k+1). Accordng to (13), compute F (x (k+1) ) and G(x (k+1) ). Step 2. If G(x (k+1) ) F (x (k+1) ) < ε, stop. Output ε-approxmaton of the domnant egenvalue of A: λ (k+1) = 1 ( ) G(x (k+1) ) + F (x (k+1) ) α, (15) 2 and the correspondng egenvector x (k+1). Otherwse, set k := k + 1 and go to Step 1. Clearly, the tensor W defned by (14) s postve and hence t s prmtve [17, Corollary 3.7]. By Theorems 2.1 and 2.2, Algorthm 5.1 s well-defned. As an mmedate consequence of Lemma 2.2, Theorem 2.4, and Theorem 5.3 n [17], we have the followng convergence theorem. Theorem 5.1 Let A be an essentally nonnegatve tensor of order m and dmensonal n, and let W be defned by (14) where ε s a suffcently small number. Then the sequences {F (x (k) )} and {G(x (k) )}, generated by Algorthm 5.1, converge to λ ε, where λ ε s the unque postve egenvalue of W. Moreover, the sequence {x (k) } converges to x ε and x ε s a postve egenvector of W correspondng to the largest egenvalue λ ε. Furthermore, lm λ ε = λ, ε 0 lm x ε = x, ε 0 where λ s the spectral radus of A + αi and x s the correspondng egenvector. In partcular, the domnant egenvalue of A s λ(a) = λ α and x s also the egenvector correspondng to λ(a). Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

10 10 L. P. ZHANG It follows from (14) that W s postve, and hence t s rreducble. Therefore, for any nonzero x P, we have Wx m 1 nt(p ), whch shows that the tensor W s prmtve. Hence, by Theorem 5.3 n [17], lm F k (x(k) ) = lm k {G(x(k) ) = λ ε, lm k x(k) = x ε. Therefore, λ ε α s an ε-approxmaton of the domnant egenvalue of A from Theorem 2.4. Furthermore, t follows from Lemma 2.2 that lm λ ε = λ, ε 0 lm x ε = x. ε 0 It s easy to see that λ α s the domnant egenvalue of A wth correspondng egenvector x. The above theorem shows that the convergence of Algorthm 5.1 s establshed for any essentally nonnegatve tensor wthout the rreducble and prmtve assumpton. In order to show the effectveness of Algorthm 5.1, we used MATLAB 7.4 to test t on the followng seven examples. The last four examples are large scale numercal examples. Example 5.1 Consder the 3-order 3-dmensonal essentally nonnegatve tensor A = [A(1, :, :), A(2, :, :), A(3, :, :)], where A(:, :, 1) = A(:, :, 2) = A(:, :, 3) = Example 5.2 Let a 3-order 3-dmensonal tensor A2 be defned by A 133 = A 233 = A 311 = A 322 = 1, A 111 = A 222 = 1 and zero otherwse. Example 5.3 Let a 3-order 4-dmensonal tensor A be defned by A 111 = A 222 = A 333 = A 444 = 1, A 112 = A 114 = A 121 = A 131 = A 212 = A 332 = A 443 = 1, and zero otherwse. Example 5.4 Let a 3-order 500-dmensonal tensor A be defned by A 1jj = 1 for j 1, A j11 = 1 for j 1, A 111 = 1, A 222 = 20, and zero otherwse. Example 5.5 Let a 4-order 100-dmensonal tensor A be defned by A 1jjj = 1 for j 1, A j111 = 1 for j 1, A 1111 = 1, A 2222 = 20, and zero otherwse. Example 5.6 Let A be a randomly generated 3-order 200-dmensonal tensor. Example 5.7 Let A be a randomly generated 3-order 50-dmensonal tensor. Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

11 THE DOMINANT EIGENVALUE OF AN ESSENTIALLY NONNEGATIVE TENSOR 11 Clearly, the essentally nonnegatve tensors defned n Examples 5.1 and 5.2 are rreducble. Whle, the essentally nonnegatve tensors defned n Examples 5.3, 5.4 and 5.5 are reducble. The tensors defned n Examples 5.6 and 5.7 are randomly generated nonnegatve tensors. The tensors defned n Examples 5.4 and 5.5 are sparse tensors. We take ε = 10 9 and termnate our teraton when one of the condtons G(x (k) ) F (x (k) ) 10 9 and k 100 s satsfed. Algorthm 5.1 produces the domnant egenvalue λ(a) = wth egenvector x = (1.0000; ; ) for Example 5.1, the domnant egenvalue λ(a) = 1 wth egenvector x = (0.5000; ; 1.000) for Example 5.2, and the domnant egenvalue λ(a) = wth egenvector x = (1.0000; ; ; ) for Example 5.3. For the large scale tensors n the last four examples, we just lst ther domnant egenvalues. Algorthm 5.1 produces the domnant egenvalue λ(a) = for Example 5.4, the domnant egenvalue λ(a) = for Example 5.5, the domnant egenvalue λ(a) = e4 for Example 5.6, and the domnant egenvalue λ(a) = e4 for Example 5.7, The detals of numercal results are reported n Tables 1 and 2. We lst the output detals at each teraton for Example 5.1 n Table 1. We also report the number of teratons (No.Iter), the elapsed CPU tme (CPU(sec)), the lower bound λ (k) = F (x (k) ) α and the upper bound λ (k) = G(x (k) ) α for k 1, the error (k) = A(x (k) ) m 1 λ (k) (x (k) ) [m 1], and the approxmaton λ (k) defned by (15) of the domnant egenvalue n Tables 1 and 2. From Tables 1 and 2, we see that the sequence generated by Algorthm 5.1 converges to the domnant egenvalue of the essentally nonnegatve tensor wthout rreducblty. Algorthm 5.1 s promsng for calculatng the domnant egenvalues of the seven examples. For the sparse tensors n Examples 5.4 and 5.5, the elapsed CPU tmes are longer because they need more teratons. Algorthm 5.1 can solve the non-sparse tensor n 20s wth the number of entres less than 100 mllon. For sparse tensors, Algorthm 5.1 s slow. Table I. Detaled output of Algorthm 5.1 for Example 5.1 k λ (k) λ (k) λ (k) λ (k) λ (k) (k) e e e e e e e e e e-9 Table II. Output of Algorthm 5.1 for Examples Example No.Iter CPU(sec) λ (k) λ (k) λ (k) λ (k) λ (k) (k) e e e e e e e e e e e e e e e e e e e e-9 Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

12 12 L. P. ZHANG 6. AN APPLICATION AND SOME CONCLUSIONS In ths paper, we have ntroduced the concepts of essentally nonnegatve tensors, whch s closely related to nonnegatve tensors. The man contrbuton s the convexty and log convexty of the domnant egenvalue of an essentally nonnegatve tensor, and hence the same for the spectral radus of a nonnegatve tensor. We also have proposed an algorthm for calculatng the domnant egenvalue and convergence analyss has been establshed for any essentally nonnegatve tensor wthout the assumptons of rreducblty and prmtveness. As an applcaton, we fnd that the convexty of the the maxmal egenvalue functon plays an mportant role n the trace-preservng problem whch arses n sgnal processng system [26, 27]. The trace-preservng problem s to determne µ(a) = mn{λ(a + D u ) : e T u = 0} and to fnd a vector u = (u 1,..., u n ) T that acheves ths mnmum, where A s an essentally nonnegatve tensor, D u s a dagonal tensor wth u 1,..., u n as the dagonal entres. By Theorem 3.1 and Corollary 3.1, ths problem s a convex problem. Motvated by the dea n [26], we guess the semsmoothness of the domnant egenvalue functon also holds and then we may propose a Newton-type algorthm to solve the trace-preservng problem. Ths s a topc n the future research. ACKNOWLEDGEMENTS The authors would lke to thank the edtor and the anonymous referees for ther constructve comments and suggestons whch lead to a sgnfcantly mproved verson of the paper. Lpng Zhang s work was supported by the Natonal Natural Scence Foundaton of Chna(Grant No ). Lqun Q s work was supported by the Hong Kong Research Grant Councl (Grant No. PolyU , , and ). Zyan Luo s work was supported by the Natonal Basc Research Program of Chna (2010CB732501) and the Natonal Natural Scence Foundaton of Chna ( ). REFERENCES 1. Loan CV. Future drectons n tensor-based computaton and modelng. Workshop Report n Arlngton, Vrgna at Natonal Scence Foundaton 2009; Q L. Bblography of egenvalues of tensors Q L. Egenvalues of a real supersymmetrc tensor. Journal of Symbolc Computaton 2005; 40: Lm LH. Sngular values and egenvalues of tensors: a varatonal approach. In Proceedngs of the IEEE Internatonal Workshop on Computatonal Advances n Mult-Sensor Addaptve Processng (CAMSAP 05) 2005; 1: IEEE Computer Socety Press, Pscataway, NJ. 5. Q L, Da HH, Han D. Condtons for strong ellptcty and M-egenvalues. Fronters of Mathematcs n Chna 2009; 4: Q L, Wang Y, Wu EX. D-egenvalues of dffuson kurtoss tensor. Journal of Computatonal and Appled Mathematcs 2008; 221: Cartwrght D, Sturmfels B. The number of egenvalues of a tensor. Lnear Algebra and Applcatons (2013); Chang KC, Pearson K, Zhang T. On egenvalue problems of real symmetrc tensors. Journal of Mathematcs Analyss and Applcatons 2009; 350: Bulò SR, Pelllo M. A generalzaton of the Motzkn-Straus theorem to hypergraphs. Optmzaton Letters 2009; 3: Bulò SR, Pelllo M. New bounds on the clque number of graphs based on spectral hypergraph theory. In: T. Stützle, ed., Learnng and Intellgent Optmzaton 2009: Sprnger Verlag, Berln. 11. Hu S, Q L. Algebrac connectvty of an even unform hypergraph. Journal of Combnatoral Optmzaton (2012); Ng M, Q L, Zhou G. Fndng the largest egenvalue of a nonnegatve tensor. SIAM Journal on Matrx Analyss and Applcatons 2009; 31: Chang KC, Pearson K, Zhang T. Perron Frobenus Theorem for nonnegatve tensors. Communcatons n Mathematcal Scences 2008; 6: Fredland S, Gaubert S, Han L. Perron-Frobenus theorem for nonnegatve multlnear forms and extensons. Lnear Algebra and Its Applcatons (2013); 438: Yang YN, Yang QZ. Further results for Perron-Frobenus Theorem for nonnegatve tensors. SIAM Journal on Matrx Analyss and Applcatons 2010; 31: Copyrght c 2013 John Wley & Sons, Ltd. Numer. Lnear Algebra Appl. (2013) Prepared usng nlaauth.cls

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