Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs
aa r X i v : . [ m a t h . SP ] O c t Finding the maximum eigenvalue of a class of tensors withapplications in copositivity test and hypergraphs
Haibin Chen ∗ , Yannan Chen † , Guoyin Li ‡ , Liqun Qi § Abstract
Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computationand numerical multilinear algebra. This paper is devoted to a semi-definite program algorithm forcomputing the maximum H -eigenvalue of a class of tensors with sign structure called W -tensors. Theclass of W -tensors extends the well-studied nonnegative tensors and essentially nonnegative tensors,and covers some important tensors arising naturally from spectral hypergraph theory. Our algorithm isbased on a new structured sums-of-squares (SOS) decomposition result for a nonnegative homogeneouspolynomial induced by a W -tensor. This SOS decomposition enables us to show that computingthe maximum H -eigenvalue of an even order symmetric W -tensor is equivalent to solving a semi-definite program, and hence can be accomplished in polynomial time. Numerical examples are givento illustrate that the proposed algorithm can be used to find maximum H -eigenvalue of an even ordersymmetric W -tensor with dimension up to 10 , H -eigenvaluesof large size Laplacian tensors of hyper-stars and hyper-trees; second, we show that the proposedSOS algorithm can be used to test the copositivity of a multivariate form associated with symmetricextended Z -tensors, whose order may be even or odd. Numerical experiments illustrate that ourstructured semi-definite program algorithm is effective and promising. Keywords: W -tensor, H -eigenvalue, sum-of-squares polynomial, hyper-star, Laplacian tensor. AMS Subject Classification(2010): ∗ School of Management Science, Qufu Normal University, Rizhao, Shandong, China. Email: [email protected] author’s work was supported by the National Natural Science Foundation of China (Grant No. 11601261). † School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China. E-mail: [email protected]. This au-thor’s work was supported by the National Natural Science Foundation of China (Grant No. 11401539) and the DevelopmentFoundation for Excellent Youth Scholars of Zhengzhou University (Grant No. 1421315070). ‡ Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia. E-mail: [email protected] author’s work was partially supported by Australian Research Council. § Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. Email:[email protected]. This author’s work was supported by the Hong Kong Research Grant Council (Grant No. PolyU501212, 501913, 15302114 and 15300715). Introduction
Finding the extremum (maximum or minimum) eigenvalue of a tensor is an important topic in tensorcomputation and numerical multilinear algebra. Various applications of extremum eigenvalues have beenfound in the current literature [6, 8, 26, 31]. For example, in [29], the sign of the minimum (maximum)eigenvalue plays a crucial role in checking the positive semi-definiteness (negative semi-definiteness) of asymmetric tensor which has applications in stability analysis of nonlinear autonomous systems involvedin automatic control. In magnetic resonance imaging [4, 32], the principal eigenvalues of an even ordersymmetric tensor associated with the fiber orientation distribution of a voxel in cerebral white matterdenote volume factions of multiple nerve fibers in this voxel. For a connected even-uniform hypergraph,it has been shown that the maximum H-eigenvalues of the Laplacian tensor and the signless Laplaciantensor are equivalent if and only if the hypergraph is odd-bipartite [18].In view of the importance of eigenvalues of tensors, many researchers have devoted themselves tothe study of numerical methods for eigenvalues of high order tensors [1, 10–12, 16, 20, 27]. In [7], Cuiet al. proposed a sophisticated Jacobian semi-definite relaxation method, which computes all of thereal eigenvalues of a small symmetric tensor. Generally speaking, it is an NP-hard problem to computeeigenvalues of a tensor even though the involved tensor is symmetric [13]. But for some tensors withspecial structures, large scale problems can be solved. In [5], an inexact curvilinear search optimizationmethod was established to compute extreme eigenvalues of Hankel tensors, whose dimension may up toone million.Nonnegative tensors is an important class of structured tensors which arises from the study of imagescience, statistics, and hypergraph theory. Ng et al. proposed an iterative method for finding the maximumH-eigenvalue of an irreducible nonnegative tensor [26]. However, the NQZ method is not always conver-gent for irreducible nonnegative tensors. Liu et al. [23] improved the NQZ method so that the refinedalgorithm is always convergent. Recently, as a more general class than nonnegative tensors, the essentiallynonnegative tensors were studied in [16, 37]. Hu et al. [16] showed that the maximum H -eigenvalue ofan even order essentially nonnegative tensor can be found by solving a polynomial optimization problem,which is equivalently reformulated as a semi-definite programming problem. On the other hand, there arealso many important classes of structured tensors which need not to be nonnegative tensors or essentiallynonnegative tensors such as the Laplacian tensor of a hypergraph. This then raise the following naturalquestion: can we compute the maximum H -eigenvalue of a given tensor with possibly negative values onthe off-diagonal elements? This is the main motivation of this paper.Next, we give a sketch of the copositivity of symmetric tensors. The definition of copositive tensorswas introduced in [30]. Recently, it has been found that copositive tensors have important applications2n vacuum stability of a general scalar potential [19], polynomial optimization [28, 35] and the tensorcomplementarity problem [2, 33, 34]. With the help of copositive tensors, Kannike [19] studied the vacuumstability of a general scalar potential of a few fields, and it is showed that how to find positivity conditionsfor more complicated potentials. Pena et al. [28] proved that recent related results for quadratic problemscan be further strengthened and generalized to higher order polynomial optimization problems over thecone of completely positive tensors or copositive tensors. Che, Qi and Wei [2] showed that the tensorcomplementarity problem defined by a strictly copositive tensor has a nonempty and compact solutionset. Song and Qi [33] proved that a real tensor is strictly semi-positive if and only if the correspondingtensor complementarity problem has a unique solution for any nonnegative vector and a real tensor issemi-positive if and only if the corresponding tensor complementarity problem has a unique solution forany positive vector. It was shown there that a real symmetric tensor is a (strictly) semi-positive tensorif and only if it is (strictly) copositive. Song and Qi [34] further presented global error bound analysisfor the tensor complementarity problem defined by a strictly semi-positive tensor. Thus, copositive andstrictly copositive tensors play an important role in the tensor complementarity problem.In this article, we propose an efficient semidefinite program algorithm to compute the maximum H -eigenvalues of even order symmetric W -tensors, which includes nonnegative tensors and essentially non-negative tensors as a subclass. This algorithm heavily relies on an important SOS representation resultfor a nonnegative polynomial induced from a W -tensor. To proceed, we first give the SOS representationresult for nonnegativity of a homogeneous polynomial induced by an even order symmetric W -tensor,which implies that the proposed algorithm maybe much more computationally efficient when the dimen-sion is large and the explicit expression of the subtensors are available. Another interesting feature ofthe W -tensor is that it also covers the Laplacian tensors of hyper-star and hyper-tree [17, 18], and hence,the maximum H -eigenvalues of even order Laplacian tensors can be efficiently computed by the proposedalgorithm. Numerical examples show that the proposed algorithm can be used to compute the maximum H -eigenvalue of some large size tensors with dimension up to 10 , Z -tensors,where the order of the tensor can be either odd or even.This paper is organized as follows. In Section 2, we recall the definitions for tensors and some basicresults about homogeneous polynomials. In Section 3, we propose an algorithm and show that the max-imum H -eigenvalue of an even order symmetric W -tensor can be computed by the proposed algorithm.In Section 4, we apply the proposed algorithm to compute the maximum H -eigenvalue of a given largesize Laplacian tensor of hyper-stars and hyper-trees. Numerical examples are also presented to show theefficiency of our method. In Section 5, we apply the approach to test the copositivity of a multivariate3orm associated with symmetric extended Z -tensors, where the order of the tensor can be either odd oreven. In Section 6, we present some final remarks and future work.Before we move on, we make some comments on notation that will be used in the sequel. Let R n be the n dimensional real Euclidean space and R n + be the set of all nonnegative vectors in R n . The set consistingof all positive integers is denoted by N . Let m, n ∈ N be two natural numbers. Denote [ n ] = { , , · · · , n } .Vectors are denoted by bold lowercase letters x , y , . . . , matrices are denoted by capital letters A, B, . . . ,and tensors are written as calligraphic capitals such as A , T , . . . . The identity tensor I with order m anddimension n is given by I i ··· i m = 1 if i = · · · = i m and I i ··· i m = 0 otherwise. The i -th unit coordinatevector in R n is denoted by e i , i ∈ [ n ]. In this section, we collect some basic definitions and facts that will be used later on. Then, we introducethe definition of W -tensors.An m -th order n -dimensional tensor A = ( a i i ··· i m ) is a multi-array of entries a i i ··· i m , where i j ∈ [ n ]for j ∈ [ m ]. If the entries a i i ··· i m are invariant under any permutation of their indices, then tensor A is called a symmetric tensor. The entries a ii ··· i , i ∈ [ n ], are the diagonal entries of A and the rest areoff-diagonal entries.We note that an m -th order n -dimensional symmetric tensor A uniquely determines an m -th degreehomogeneous polynomial f A ( x ) on R n : for all x = ( x , · · · , x n ) T ∈ R n , f A ( x ) = A x m = X i ,i , ··· ,i m ∈ [ n ] a i i ··· i m x i x i · · · x i m . (2.1)Conversely, an m -th degree homogeneous polynomial function f A ( x ) on R n also uniquely corresponds to asymmetric tensor. Furthermore, an even order tensor A is called positive semi-definite (positive definite)if f A ( x ) ≥ f A ( x ) > for all x ∈ R n \{ } . Recall that for a polynomial f on R n , we say f is a sums-of-squares (SOS) polynomial if there exist r ∈ N and polynomials f i , i = 1 , . . . , r such that f = P ri =1 f i . Suppose that m is even. We denote theset consisting of all SOS polynomials of degree m by Σ m [ x ]. In (2.1), if f A ( x ) is an SOS polynomial,then we say tensor A has an SOS tensor decomposition [3, 15]. It is clear that a tensor with SOS tensordecomposition must be a positive semi-definite tensor, but not vice versa. For all x ∈ R n , considera homogeneous polynomial f ( x ) = P α f α x α with degree m , where α = ( α , · · · , α n ) ∈ ( N ∪ { } ) n , x α = x α · · · x α n n and | α | := P ni =1 α i = m . Let f m,i be the coefficient associated with x mi . Let e i be the4 -th unit vector and letΩ f = { α = ( α , · · · , α n ) ∈ ( N ∪ { } ) n : f α = 0 and α = m e i , i = 1 , · · · , n } . Then, f can always be written as f ( x ) = n X i =1 f m,i x mi + X α ∈ Ω f f α x α . We now recall the definitions of eigenvalues and eigenvectors for a tensor [22, 29].
Definition 2.1
Let C be the complex field. Let A = ( a i i ··· i m ) be a symmetric tensor with order m dimension n . A pair ( λ, x ) ∈ C × ( C n \ { } ) is called an eigenvalue-eigenvector pair of tensor A , if theysatisfy A x m − = λ x [ m − , where A x m − and x [ m − are all n dimensional column vectors given by A x m − = n X i , ··· ,i m =1 a ii ··· i m x i · · · x i m ≤ i ≤ n and x [ m − = ( x m − , . . . , x m − n ) T ∈ C n . If the eigenvalue λ and the eigenvector x are real, then λ is called an H -eigenvalue of A and x is itscorresponding H -eigenvector [29]. An important fact is that an even order symmetric tensor is positivesemi-definite (definite) if and only if all H -eigenvalues of the tensor are nonnegative (positive). It shouldbe noted that even order symmetric tensors always have H -eigenvalues.The following lemma will play an important role in our later analysis [29]. Lemma 2.1
Let A be a symmetric tensor with order m and dimension n , where m is even. Denote theminimum H -eigenvalue and maximum H -eigenvalue of A by λ min ( A ) and λ max ( A ) respectively. Then,we have λ min ( A ) = min x = A x m k x k mm = min k x k m =1 A x m , λ max ( A ) = max x = A x m k x k mm = max k x k m =1 A x m , where k x k m = ( P ni =1 | x i | m ) m . Now, we are ready to define W -tensors formally. For I ⊆ [ n ], we denote by x I the set of variables { x i : i ∈ I } and by R [ x I ] the polynomial ring in these variables. For a set S with finitely many members,we use | S | to denote its cardinality. 5 efinition 2.2 Let A = ( a i i ··· i m ) be a tensor with order m and dimension n . We say A is a W -tensorif there exist s ∈ N with s ≤ n and index sets Γ l ⊆ [ n ] , l ∈ [ s ] with S sl =1 Γ l = [ n ] and Γ l = Γ l for all l = l such that (i) either s = 1 or | (cid:16)S p − l =1 Γ l (cid:17) T Γ p | ≤ for all ≤ p ≤ s . (ii) A x m = P sl =1 A Γ l x m Γ l for all x ∈ R n , where for each l ∈ [ s ] , A Γ l is a tensor with order m anddimension | Γ l | . (iii) for each l ∈ [ s ] , the tensor A Γ l satisfies either one of the following two conditions: (1) there exists { i , i , · · · , i m } ∈ Γ l with ( i , i , · · · , i m ) = ( i, · · · , i ) , i ∈ [ n ] such that the off-diagonalentries of A Γ l equal zero for all ( i , i , · · · , i m ) ∈ Γ l \{ π ( i i · · · i m ) } , where π ( i i · · · i m ) denotesall the permutation of ( i i · · · i m ) . (2) all the off-diagonal entries of A Γ l are nonnegative. Recall that a tensor A is called an essentially nonnegative tensor if its off-diagonal entries, A i ,i , ··· ,i m with { i , i , . . . , i m } / ∈ { ( i, i, . . . , i ) : 1 ≤ i ≤ n } , are all nonnegative. It is obvious that essentially nonnegativetensors are W -tensors, and the converse is not true in general. Moreover, it is easy to check that condition( i ) is automatically satisfied if Γ , · · · , Γ s are disjoint i.e., Γ l T Γ l = ∅ for all l = l . Finally, as wewill see later in Section 4, the significance of the W -tensors is that it not only extends the essentiallynonnegative tensors but also covers important structured tensors which naturally arises in the hypergraphtheory. Remark 2.1
From Definition 2.2, it can be verified that if A is a W -tensor and D is a diagonal tensor,then D + A is also a W -tensor. H -eigenvalue of a symmetric W -tensor In this section, we show that the maximum H -eigenvalue of an even order symmetric W -tensor can becomputed by solving a semi-definite programming problem, and so, can be accomplished in polynomialtime. We first recall a useful lemma, which provides us a simple criterion for determining whether ahomogeneous polynomial with only one mixed term is a sum-of-squares polynomial or not [9]. Lemma 3.1
Assume b , b , · · · , b n ≥ and d ∈ N . Let a , a , · · · , a n ∈ N and P ni =1 a i = 2 d . Considerthe homogeneous polynomial f ( x ) defined by f ( x ) = b x d + · · · + b n x dn − µx a · · · x a n n . Let µ = 2 d Q a i =0 , ≤ i ≤ n ( b i a i ) ai d . Then, the following statements are equivalent: f is a nonnegative polynomial i.e., f ( x ) ≥ for all x ∈ R n ; (ii) f is an SOS polynomial. To proceed, we need the following SOS representation result for nonnegativity of a polynomial inducedby a symmetric W -tensor. Theorem 3.1
Let A be a symmetric W -tensor with even order m and dimension n and let f ( x ) = −A x m .Let A Γ l and Γ l , l ∈ [ s ] be defined as in Definition 2.2. Suppose that f ( x ) ≥ for all x ∈ R n . Then, thereexist h l ∈ R [ x Γ l ] , l ∈ [ s ] and ρ li ∈ R , i ∈ [ n ] and l ∈ [ s ] , such that each h l is a sum-of-squares polynomialwith h l ( x Γ l ) = −A Γ l x m Γ l + P i ∈ Γ l ρ li x mi , for each i = 1 , . . . , n , X l ∈ Λ( i ) ρ li = 0 , with Λ( i ) = { ≤ l ≤ s : i ∈ Γ l } , and f ( x ) = h ( x Γ ) + · · · + h s ( x Γ s ) . Proof. As A is a symmetric W -tensor, there exist s ∈ N with s ≤ n and index sets Γ l ⊆ [ n ], l ∈ [ s ]with S sl =1 Γ l = [ n ] and Γ l = Γ l for all l = l , such that conditions (i)-(iii) hold in Definition 2.2. Inparticular, condition (ii) shows that f = f + · · · + f s with f l ∈ R [ x Γ l ], l ∈ [ s ], be homogeneous polynomialswith degree m given by f l ( x Γ l ) = −A Γ l x m Γ l . Let us prove the conclusion of this theorem by induction on s . (1) We first prove the trivial case, i.e. s = 1. Then condition (iii) of Definition 2.2 implies that f ( x )is either a polynomial with only one mixed term or f ( x ) = −A x m = − n X i =1 a ii ··· i x mi − X δ i i ··· im =0 a i i ··· i m x i · · · x i m , (3.1)where δ i i ··· i m equals one if i = i = · · · = i m , and equals zero otherwise; and a i i ··· i m ≥ i , i , · · · , i m ∈ [ n ] with δ i i ··· i m = 0. If the first case holds, we obtain that f is SOS from Lemma 3.1since f ( x ) ≥ x ∈ R n . If (3.1) is true, then A is an even order essentially nonnegative tensor. Itfollows from [16, Proposition 3.1] that f is SOS (see also [15]), and hence, the desired result holds in thecase s = 1. (2) [ Initial Step ] Let 1 < s ≤ n . We start with s = 2. Then, it holds that f = f + f where f ( x Γ ) = −A Γ x m Γ and f ( x Γ ) = −A Γ x m Γ . From condition (i) of Definition 2.2, we see that there exists i ∈ [ n ] such that Γ T Γ ⊆ { i } .7f Γ T Γ = ∅ , it can be easily verified that f l ≥ , l ∈ { , } since f ( x ) ≥
0. So, condition (iii) ofDefinition 2.2 implies that each f l is either a homogeneous polynomial with only one mixed term or ahomogeneous polynomial such that f l ( x Γ ) = −A l ( x Γ ) m and A l is an even order essentially nonnegativetensor. This means that f , f are sum-of-squares polynomials. Thus, f = f + f is SOS and the desiredresult follows with h = f , h = f .If Γ T Γ = { i } , denote b Γ l = Γ l \{ i } , l ∈ { , } . Without loss of generality, we assume b Γ l = ∅ , l ∈{ , } and we order Γ l , l = 1 ,
2, in such a way that x Γ = ( x c Γ , x i ) and x Γ = ( x i , x c Γ ). We first see that α = inf { f ( a ,
1) : a ∈ R | c Γ | } > −∞ . (3.2)Otherwise, there exists a k ∈ c Γ such that f ( a k , → −∞ . Let c Γ ∈ R | c Γ | be the vector such that all itsentries equal 1. Note that ( a k , , c Γ ) ∈ R n , and so0 ≤ f ( a k , , c Γ ) = f ( a k ,
1) + f (1 , c Γ ) → −∞ , which is impossible, and hence, (3.2) is true. Let q ( x ) = αx mi . Define h = f − q and h = f + q . Wenow verify that h = f − q ≥ over R | Γ | and h = f + q ≥ over R | Γ | . To see this, take any ( a , ∈ R | Γ | with a ∈ R | c Γ | , and so h ( a ,
1) = f ( a , − α = f ( a , − inf { f ( a ,
1) : a ∈ R | c Γ | } ≥ . Noting that h is a homogeneous polynomial with an even order degree m , it follows that h ( x Γ ) ≥ for all x Γ = ( a , s ) ∈ R | c Γ | × R with s = 0 . Then, by the continuity of h , it shows that h ≥ R | Γ | . Moreover, take any (1 , b ) ∈ R | Γ | with b ∈ R | c Γ | . Fix an arbitrary ǫ >
0. Take z ǫ ∈ R | c Γ | be such that f ( z ǫ , ≤ inf { f ( x | c Γ | ,
1) : x | c Γ | ∈ R | c Γ | } + ǫ .Then, h (1 , b ) = f (1 , b ) + inf { f ( x | c Γ | ,
1) : x | c Γ | ∈ R | c Γ | }≥ f (1 , b ) + f ( z ǫ , − ǫ = f ( z ǫ , , b ) − ǫ ≥ − ǫ, where the last inequality follows by the fact that f ( x ) ≥ x ∈ R n . Letting ǫ →
0, we have h (1 , b ) ≥ b ∈ R | c Γ | . Similarly, using the fact that h is a continuous homogeneous polynomialwith even degree m , we see that h ≥ R | Γ | .To finish the proof of the initial step i.e. s = 2, it remains to observe from condition (iii) of Definition2.2 that each h l ∈ R [ x Γ l ] is either a polynomial with only one mixed term or a homogeneous polynomial8uch that h l ( x Γ l ) = −H l x Γ l m and H l is an essentially nonnegative tensor. Similar to the analysis of (1) ,we know that h l , l ∈ { , } are SOS. Therefore, the conclusion holds with s = 2. (3) [ Induction Step ] Suppose that the conclusion is true for s = p −
1. We now examine the casewith s = p . In this case, we have f = f + f + · · · + f p = ˆ f + f p , where ˆ f = f + f + · · · + f p − . Then, ˆ f is a homogeneous polynomial with ˆ f ∈ R [ x S p − l =1 Γ l ]. By thedefinition of W -tensor, there exists i p ∈ [ n ] such that p − [ l =1 Γ l ! \ Γ p ⊆ { i p } . Similarly to the proof in initial step (2) , there exist ˆ h ∈ R [ x S p − l =1 Γ l ], h p ∈ R [ x Γ p ] and finite number ρ ∈ R such that ˆ h = ˆ f − ρx mi p ≥ over R | S p − l =1 Γ l | , h p = f p + ρx mi p ≥ over R | Γ p | . By the induction hypothesis that the conclusion holds when s = p −
1, we know that ˆ h = h + h + · · · + h p − ,where h l ∈ R [ x Γ l ], l ∈ [ p − h l is a sum-of-squares polynomial with h l ( x Γ l ) = A Γ l x m Γ l + P i ∈ Γ l ρ li x mi for some ρ li ∈ R satisfying P p − l =1 P i ∈ Γ l ρ li = 0 andˆ f ( x ) = h ( x Γ ) + · · · + h p − ( x Γ p − ) . On the other hand, h p = f p + ρx mi p is either a homogeneous polynomial with one mixed term or a ho-mogeneous polynomial a homogeneous polynomial such that h p ( x ) = −H p x m and H p is an essentiallynonnegative tensor. Thus h p is SOS and the desired results hold. (cid:3) Theorem 3.2
Let A be a symmetric W -tensor with even order m and dimension n . Then, it holds that λ max ( A ) = min t ∈ R ,µ ∈ R { t | t − A x m + µ ( k x k mm − ∈ Σ m [ x ] } . (3.3) Moreover, let A Γ l , l ∈ [ s ] be subtensors of A as defined in Definition 2.2, then we also have λ max ( A ) = min t,ρ li ∈ R { t : −A Γ l x m Γ l + X i ∈ Γ l ρ li x mi ∈ Σ m [ x Γ l ] , l ∈ [ s ] , X l ∈ Λ( i ) ρ li ≤ t, i ∈ [ n ] } , (3.4) where, for each i = 1 , . . . , n, Λ( i ) = { ≤ l ≤ s : i ∈ Γ l } .Proof. Let t ∗ = λ max ( A ). By Lemma 2.1, we have that f ( x ) = t ∗ m X i =1 x mi − A x m ≥ , for all x ∈ R n . B = A − t ∗ I . It can be easily verified that f ( x ) = −B x m ≥ x ∈ R n . By Definition 2.2and Remark 2.1, we know that B is a symmetric W -tensor since A is W -tensor. Let B Γ l , l ∈ [ s ] denotethe subtensors of B . So, it is easy to verify that B Γ l only differs A Γ l by a diagonal tensor on R | Γ l | . ByTheorem 3.1, there exist sum-of-squares polynomials h l ∈ R [ x Γ l ] , l ∈ [ s ] such that f = h + h + · · · + h s (3.5)and h l ( x Γ l ) = −A Γ l x m Γ l + X i ∈ Γ l ¯ ρ li x mi (3.6)for some ¯ ρ li ∈ R . Thus, f ( x ) = −B x m = −A x m + t ∗ k x k mm is an SOS polynomial, and so, t = µ = t ∗ isfeasible for the problem min t ∈ R ,µ ∈ R { t | t − A x m + µ ( k x k mm − ∈ Σ m [ x ] } . So, it follows that min t ∈ R ,µ ∈ R { t | t − A x m + µ ( k x k mm − ∈ Σ m [ x ] } ≤ t ∗ = λ max ( A ) . On the other hand, for all x ∈ R n , take any ( t, µ ) with t − A x m + µ ( k x k mm − ∈ Σ m [ x ] . Then, it holdsthat t − A x m + µ ( k x k mm − ≥ , ∀ x ∈ R n , which implies that t ≥ µ and µ k x k mm ≥ A x m for all x ∈ R n . Thus, we know that t ∗ ≥ A x m for all x ∈ R n with k x k mm = 1. Therefore, (3.3) follows.To see the second part, by (3.5) and (3.6), we see that X l ∈ Λ( i ) ¯ ρ li = t ∗ . Combining this with the fact that each h l is a sum-of-squares polynomial, gives us that t = t ∗ and ρ li = ¯ ρ li is feasible for min t,ρ li ∈ R { t : −A Γ l x m Γ l + X i ∈ Γ l ρ li x mi ∈ Σ m [ x Γ l ] , l ∈ [ s ] , X l ∈ Λ( i ) ρ li ≤ t, i ∈ [ n ] } , which implies thatmin t,ρ li ∈ R { t : −A Γ l x m Γ l + X i ∈ Γ l ρ li x mi ∈ Σ m [ x Γ l ] , l ∈ [ s ] , X l ∈ Λ( i ) ρ li ≤ t, i ∈ [ n ] } ≤ t ∗ . Moreover, by a direct computation, the reverse inequality always hold such thatmin t,ρ li ∈ R { t : −A Γ l x m Γ l + X i ∈ Γ l ρ li x mi ∈ Σ m [ x Γ l ] , l ∈ [ s ] , X l ∈ Λ( i ) ρ li ≤ t, i ∈ [ n ] } ≥ t ∗ . Therefore, the desired results hold. (cid:3) emark 3.1 (Polynomial time solvability of maximum H -eigenvalue of the symmetric W -tensor) As explained in [15], checking a sum-of-squares polynomial can be equivalently rewritten as asemi-definite programming problem. Then, Theorem 3.2 shows that if A is a symmetric W -tensor witheven order, its maximum H -eigenvalue can be found by solving a semi-definite problem, and so can bevalidated in polynomial time. According to the equality (3.3), we obtain a basic algorithms for finding the maximum eigenvalue of a W -tensor. Algorithm 3.1
Given m, n ∈ N and let m be even. Input an m -th order n -dimensional W -tensor A = ( a i i ··· i m ) . Let f A ( x ) = A x m and g ( x ) = k x k mm − P i ∈ [ n ] x mi − . Solve the following optimization problem min t ∈ R ,µ ∈ R { t | t − f A ( x ) + µg ( x ) ∈ Σ m [ x ] } . Output t . When the subtensors A l of a W -tensor A can be explicitly exploited, we can have the following refinedalgorithm for computing the maximum eigenvalue of a W -tensor. Algorithm 3.2
Given m, n ∈ N and let m be even. Input an m -th order n -dimensional W -tensor A = ( a i i ··· i m ) with its subtensors A Γ l , l ∈ [ s ] asdefined in Definition 2.2. Solve the following optimization problem min t,ρ li ∈ R { t : −A Γ l x m Γ l + X i ∈ Γ l ρ li x mi ∈ Σ m [ x Γ l ] , l ∈ [ s ] , X l ∈ Λ( i ) ρ li ≤ t, i ∈ [ n ] } , where Λ( i ) = { ≤ l ≤ s : i ∈ Γ l } . Output t . Remark 3.2 (Comparison with Algorithms 3.1 and 3.2)
It is known that checking a polynomial witheven degree d and dimension n is sums-of-squares or not can be equivalently reformulated as a semi-definiteprogramming problem which can be done via the Matlab Toolbox YALMIP [24, 25]. Thus, Algorithms 3.1and 3.2 can both be used to compute the maximum H -eigenvalue of a W -tensor via YALMIP and thecommonly used SDP solver such as SeDuMi [36]. Algorithm 3.2 requires the explicit expression of thesubtensors of a W -tensor while Algorithm 3.1 does not need this information. On the other hand, in thecase where n is large and the explicit expression of the subtensors are available, Algorithm 3.2 can bemuch more computationally efficient. Indeed, note that checking whether a polynomial with even degree d nd dimension n is sums-of-squares or not leads to a semi-definite programming problem whose size (themaximum of the number of the variables and the number of involved linear constraints) is (cid:0) n + dd (cid:1) . So, for an m -th order n -dimensional tensor, Algorithm 3.1 amounts of solving a semi-definite programming problemwith size (cid:0) n + mm (cid:1) ; while Algorithm 3.2 leads to a semi-definite programming problem with size s (cid:0) k + mm (cid:1) where k = max {| Γ l | : 1 ≤ l ≤ s } , which is much smaller than (cid:0) n + mm (cid:1) if n is large and k is small. For example, if s = n/ , k = 4 and m = 4 , then (cid:0) n + mm (cid:1) is of the order n ; while s (cid:0) k + m m (cid:1) is of the order n . Next, we present an example to illustrate that Algorithm 3.2 can be used to compute the maximum H -eigenvalue for large size W-tensors (dimension up to 10 , Example 3.1
Let n = 4 k with k ∈ N . Consider the symmetric tensor A with order and dimension n where A = A = · · · = A nnnn = n, A i i i i = − , for all ( i , i , , i , i ) = π (4 l − , l − , l − , l ) , l = 1 , · · · , n , and A i i i i = 0 otherwise. Here π ( i , · · · , i ) denotes all the possible permutation of ( i , · · · , i ) . Clearly A is not a nonnegative tensor (or an essentially nonnegative tensor). The tensor A corresponds to aunique homogeneous polynomial f A ( x ) = A x m = n ( x + · · · + x n ) − n/ X l =1 x l − x l − x l − x l . Let Γ l = (4 l − , l − , l − , l ) , l = 1 , · · · , n . Then, by Definition 2.2, A is a W -tensor with subtensors A Γ l , l = 1 , . . . , n , defined by A Γ l x l = n ( x l − + x l − + x l − + x l ) − x l − x l − x l − x l . Moreover, using geometric mean inequality, we can directly verify that the true maximum H -eigenvalue of A is λ H max ( A ) = n + 1 .We compute the maximum H -eigenvalue of A using Algorithm 3.2 for the cases of n = 500 to , ,where True λ H max ( A ) and Est. λ H max ( A ) denote the true maximum H -eigenvalue and estimated maximum H -eigenvalue respectively. The results are summarized in Table 1. Obviously, Algorithm 3.2 finds themaximum H -eigenvalue of A exactly. The CPU-time (measured in seconds) for converting the sums-of-squares problem to SDP via the Matlab toolbox YALMIP and solving SDP via the commonly used SDPsoftware SeDuMi [36] is reported in columns YALMIP and SeDuMi, respectively. When the dimension ofthe tensor increases, the CPU-time for YALMIP and SeDuMi grows steadily. m n True λ H max ( A ) Est. λ H max ( A ) YALMIP SeDuMi4 500 501 501.0000 10.2 4.34 1000 1001 1001.0000 29.9 12.64 2000 2001 2001.0000 112.5 26.34 5000 5001 5001.0000 650.8 66.44 10000 10001 10001.0000 2164.8 136.7 Throughout this part, unless stated otherwise, a hypergraph means an undirected simple k -uniform hy-pergraph G = ( V, E ), where E ⊆ V . The elements of V = V ( G ), which is labeled as [ n ] = { , , · · · , n } ,are referred to as vertices and the elements of E = E ( G ) are called edges. Recall that a simple hypergraphis a hypergraph where none of its edges is contained within another. We say a hypergraph is m -uniformif for every edge e ∈ E , it holds that | e | = m . For a subset S ⊆ [ n ], we denote by E S the set of edges { e ∈ E | S ∩ e = ∅} . For a vertex i ∈ V , we simplify E { i } as E i . The cardinality of the set E i is definedas the degree of the vertex i , which is denoted by d i .Now, we first introduce several basic definitions that will be studied. The following definition forLaplacian tensor and signless Laplacian tensor were proposed by Qi in [31]. For other related definitionssee [14, 21]. Definition 4.1 (Laplacian and signless Laplacian of hypergraphs)
Let G = ( V, E ) be an m -uniformhypergraph where V = { , , · · · , n } . The adjacency tensor of G is defined as the m th order n dimensionaltensor A with a i i ··· i m = m − { i , i , · · · , i m } ∈ E, otherwise . Let D be an m th order n dimensional diagonal tensor with its diagonal elements equal to the degree ofvertex i , for all i ∈ [ n ] . Then L = D − A is the Laplacian tensor of hypergraph G , and Q = D + A is thesignless Laplacian tensor of hypergraph G . It can be easily verified that the signless Laplacian tensor of a hypergraph is a nonnegative tensor, andso, is, in particular, a W -tensor. On the other hand, the off-diagonal elements of the Laplacian tensorof a hypergraph can be negative. Below, we show that the Laplacian tensors of two important types ofhypergraphs are indeed W -tensors, and hence, their maximum H -eigenvalue can be found in polynomialtime via Algorithm 3.2. 13 .1 Laplacian tensors of hyper-stars To move on, we first recall the concept of a hyper-star [18].
Definition 4.2
Let V = { , , · · · , n } and E is a set of subsets of V . Let G = ( V, E ) be a m -uniformhypergraph. If there is a disjoint partition of the vertex set V as V = V ∪ V ∪ · · · ∪ V s such that | V | = 1 and | V | = | V | = · · · = | V s | = m − , and E = { V ∪ V i | i ∈ [ s ] } , then G is called a hyper-star . It is an immediate fact that, with a possible renumbering of the vertices, all the hyper-stars with the samesize are identical.
Theorem 4.1
Let G = ( V, E ) be a hyper-star. Then its Laplacian tensor is a symmetric W -tensor.Proof. Let A and L be the adjacent tensor and Laplacian tensor of G respectively. Then L = D − A ,where D is the diagonal tensor with its diagonal entries d i i.e., the degree of the vertex i ∈ [ n ]. Assume V = [ n ] and | E | = s . Let V = { i } and V l = { i l , i l , · · · , i lm − } , l ∈ [ s ]. Define Γ l = { i , i l , i l , · · · , i lm − } , l ∈ [ s ]. It holds that (cid:16)S p − l =1 Γ l (cid:17) T Γ p = { i } , for all 2 ≤ p ≤ s . By Definition 4.1 and Definition 4.2, weknow that L i i ··· i m = d i if i = i = · · · = i m = i − m − if { i , i , · · · , i m } = V ∪ V l for some l ∈ [ s ]0 otherwise and L x m = m X i =1 d i x mi − X { i , ··· ,i m }∈ E m − x i x i · · · x i m . For any l ∈ [ s ], define m th order | Γ l | dimensional tensors L Γ l such that( L Γ l ) i i ··· i m = d i s if i = i = · · · = i m = i d i lj if i = i = · · · = i m = i lj , j = 1 , , · · · , m − − m − if { i , i , · · · , i m } = V ∪ V l otherwise . It can be verified that L x m = P sl =1 L Γ l x m Γ l for all x ∈ R n , and L is a symmetric W -tensor from Definition2.2. (cid:3) The notion of hyper-tree is defined based on a normal graph. Let G = ( V, E ) be a usual graph, that is a2-uniform hypergraph. If any two vertices of G are connected by exactly one path, then G is called a tree.14n particular, a tree is called a rooted oriented tree if one vertex has been designated the root, in whichcase the edges have a natural orientation which one orders the edge from the top of the rooted tree (root)to the bottom and from the left to the right.According to [17], an m -uniform hyper-tree G = ( V, E ) is the m th-power of a tree such that there existsa tree T = ( V , E ) and additional vertices ¯ V = { i e, , i e, , · · · , i e,m − | e ∈ E } satisfying V = V ∪ ¯ V . Ifthe vertices of ¯ V are all distinct and the tree is an oriented rooted tree, we call it an m -uniform orientedrooted hyper-tree generated by independent vertices . As an illustration, the following hypergraph V = { , . . . , } with E = { (1 , , , , (4 , , , , (4 , , , , (1 , , , , (13 , , , , (16 , , , } is a 4-uniform hyper-tree generated by independent vertices, because it can be formed by the tree T =( V , E ), where V = { , , , , , , } , E = { (1 , , (4 , , (4 , , (1 , , (13 , , (16 , } and the additional vertices ¯ V = { , , , , , , , , , , , } . Theorem 4.2
Let G = ( V, E ) be an m -uniform oriented rooted hyper-tree generated by independent ver-tices. Then, its Laplacian tensor is a symmetric W -tensor.Proof. Let the adjacent tensor and Laplacian tensor of G be denoted by A and L respectively. Let D be the diagonal tensor with its diagonal entries d i i.e. the degree of the vertex i ∈ [ n ]. So, it holds that L = D − A . From Remark 2.1, it suffices to show that −A is a W -tensor. As G = ( V, E ) is an m -uniformoriented rooted hyper-tree by independent vertices, there exists a oriented tree T = ( V , E ) and additionaldistinct vertices ¯ V = { i e, , i e, , · · · , i e,m − | e ∈ E } satisfying V = V ∪ ¯ V . Without loss of generality,suppose V = { , , · · · , n } and E = { e , e , · · · , e s } , where s = | E | . Let ¯Γ l = { i ∈ V | i ∈ e l } , l ∈ [ s ]. Then, it satisfies that V = S sl =1 ¯Γ l and | (cid:16)S p − l =1 ¯Γ l (cid:17) T ¯Γ p | ≤ ≤ p ≤ s . Now, define Γ l = ¯Γ l S { i e l , , i e l , , · · · , i e l ,m − } , l ∈ [ s ]. Then, we see that each Γ l correspondsan edge of the hyper-tree G , and it can be easily verified that all conditions (i)-(iii) of Definition 2.2 aresatisfied. Hence −A is a W -tensor, and it follows that the Laplacian tensor L is a W -tensor. (cid:3) Throughout this section, all numerical experiments are performed on a desktop, with 3.47 GHz quad-coreIntel E5620 Xeon 64-bit CPUs and 4 GB RAM, equipped with Matlab 2015.15 xample 4.1
Let G = ( V, E ) be a 4-uniform hyper-star. Suppose E = { e , e , · · · , e k } , where k ∈ N .Then, it holds that | V | = 3 k + 1 . Without loss of generality, assume e j = { , j − , j, j + 1 } , j ∈ [ k ] . So, the Laplacian tensor L of the hyper-star are 4th order and (3 k + 1) -dimensional tensors such that L i i i i = k if i = i = i = i = 1 , if i = i = i = i = i, i ∈ { , , · · · , k + 1 } , − if ( i , i , i , i ) = (1 , j − , j, j + 1) , for some j ∈ [ k ] , otherwise . So, the homogeneous polynomials corresponding to L is L x = kx + x + · · · + x n − k X j =1 x x j − x j x j +1 . We now compute the maximum H -eigenvalues of L using Algorithm 3.2. Table 2: Test results for Laplacian tensor of Example 4.1 m k n
True λ H max ( L ) Est. λ H max ( L ) YALMIP SeDuMi4 10 31 10.0137 10.0137 10.1 3.04 100 301 100.0001 100.0001 97.4 5.04 500 1501 500.0000 500.0000 557.9 28.64 1000 3001 1000.0000 1000.0000 1112.1 64.04 2000 6001 2000.0000 2000.0000 1907.2 205.1 Using Algorithm 3.2, we compute the maximum H-eigenvalue of the Laplacian tensor λ H max ( L ) of four-uniform hyper-stars with edges ranging from ten to two thousand. For each case, we obtain λ H max ( L ) within minutes. It is known that the true maximum H -eigenvalue λ H max ( L ) is the unique root of the polynomialequation (1 − x ) m − ( x − k ) + k = 0 in the open interval ( k, k + 1) , where k is the number of edges and m is the degree of the hypergraph. The preceding polynomial equation is solved by the Matlab command“ vpnsolve ”. The results are summarized in Table 2 where the meanings of the data are the same as inTable 1. It can be easily seen that the maximum H-eigenvalue estimated by Algorithm 3.2 are consistentwith the true maximum H-eigenvalues. Example 4.2
Let k, m ∈ N . Suppose m is an even number. Assume G = ( V, E ) is an m -uniformhypergraph with vertices and edges such that V = { , , · · · , k ( m −
1) + 1 } , E = { e , e , · · · , e k } , here each edge e l = { ( l − m −
1) + 1 , ( l − m −
1) + 2 , · · · , l ( m −
1) + 1 } , l ∈ [ k ] . So, the hypergraph G = ( V, E ) is the case of a hyper-tree with its Laplacian tensor L is an m th order n = k ( m −
1) + 1 dimensional tensor. By a direct computation, we obtain L i i ··· i m = if i = i = · · · = i m = l ( m −
1) + 1 , l ∈ { , , · · · , k − } , if i = i = · · · = i m = i, i ∈ [ n ] \{ ( m −
1) + 1 , m −
1) + 1 , · · · , ( k − m −
1) + 1 } , − m − if { i , i , · · · , i m } = e l , for some l ∈ [ k ] , otherwise . Moreover, we have that L x m = X i ∈ [ n ] \{ l ( m − | l ∈ [ k − } x mi + 2 k − X l =1 x ml ( m − − m k X l =1 x ( l − m − x ( l − m − · · · x l ( m − . Combining this with Algorithm 3.2, we compute the maximum H -eigenvalues of L with different m and n ,and the results are listed in the Table 3. It should be noted that Hu et al. [18] showed that the maximumH-eigenvalues of the Laplacian tensor L and the signless Laplacian tensor Q are equivalent when theeven-uniform hypergraph is connected and odd-bipartite. On the other hand, since the signless Laplaciantensor is nonnegative, its maximum H -eigenvalue could be computed by the classical NQZ algorithm [26].Direct verification shows that the hypergraph discussed in this example is connected and odd-bipartite, andhence, the maximum H -eigenvalue of its Laplacian tensor L can also be computed by the NQZ method. Tocompare the performance of our method and the NQZ method, we list the estimated maximum H-eigenvalueof the signless Laplacian tensor λ H max ( Q ) as well as CPU-time of the NQZ algorithm in Table 3. It can beseen that estimated maximum H-eigenvalues of λ H max ( L ) and λ H max ( Q ) coincide which numerically verifiesthe assertion in [18]. Interestingly, one also observes that Algorithm 3.2 is indeed faster than the first-orderNQZ algorithm [26] as it exploits the structure of the underlying problem. Table 3: Test results for Laplacian tensor of Example 4.2 m k n
Est. λ H max ( L ) YALMIP SeDuMi Est. λ H max ( Q ) NQZ4 100 301 2.9997 4.4 3.3 2.9997 9.44 400 1201 3.0000 28.4 12.7 3.0000 337.04 1000 3001 3.0000 160.6 36.4 3.0000 2678.36 100 501 2.6954 19.9 14.2 2.6954 24.36 400 2001 2.6956 282.1 61.7 2.6956 923.96 1000 5001 2.6956 2828.3 144.2 2.6956 6959.417 Applications in copositivity test of tensors
In this section, we present further applications on testing the copositivity of a multivariate form associatedwith symmetric extended Z-tensors, where the order of the tensor can be either odd or even. We firstrecall the definition of extended Z -tensors [3]. Definition 5.1
A symmetric tensor A is called an extended Z -tensor if its associated polynomial f A ( x ) = A x m satisfies that there exist s ∈ N with s ≤ n and index sets Γ l ⊆ { , · · · , n } , l = 1 , · · · , s with S sl =1 Γ l = { , · · · , n } and Γ l ∩ Γ l = ∅ for all l = l such that f ( x ) = n X i =1 f m,i x mi + s X l =1 X α l ∈ Ω l f α l x α l , where Ω l = { α ∈ ([ n ] ∪ { } ) n : | α | = m, x α = x i x i · · · x i m , { i , · · · , i m } ⊆ Γ l , and α = m e i , i = 1 , · · · , n } for each l = 1 , · · · , s and either one of the following two conditions holds: (1) f α l = 0 for all but one α l ∈ Ω l ; (2) f α l ≤ for all α l ∈ Ω l . Let A = ( a i i ··· i m ) be a symmetric tensor with order m and dimension n . Then A is copositive if andonly if A x m = n X i ,...,i m =1 a i ...i m x i · · · x i m ≥ , ∀ x ∈ R n + , which is equivalent to h ( x ) = A h x m = n X i ,...,i m =1 a i ...i m x i · · · x i m ≥ , ∀ x ∈ R n , (5.1)where A h is a symmetric tensor with order 2 m and dimension n .In particular, if A is a symmetric extended Z -tensor (odd or even order), then A h is also an even orderextended Z -tensor [3]. Thus, −A h is an even order W -tensor. Let f ( x ) = −A h x m . Then, by Theorem3.2 and (5.1), we have the following corollary. Corollary 5.1
Let A be a symmetric extended Z -tensor with order m and dimension n . For x ∈ R n ,suppose A h and f ( x ) are defined as above. Then, A is copositive if and only if min t ∈ R { t | t k x k m m − f ( x ) ∈ Σ m [ x ] } ≤ . (5.2)We now use the above corollary to test the copositivity of symmetric extended Z -tensors with order m and dimension n . The concrete process is listed below.18 rocedure (i) Given ( m, n, s, k, M ) with n = sk , where n and m are the dimension and the order of the randomlygenerated tensor, respectively, and M is a large positive constant.(ii) Randomly generate a partition of the index set { , · · · , n } , { Γ , · · · , Γ s } , such that | Γ i | = k , i =1 , · · · , s and Γ i ∩ Γ i ′ = ∅ for all i = i ′ . For each i = 1 , · · · , s −
1, generate a random multi-index( l i , · · · , l im ) with l ij ∈ Γ i , j = 1 , · · · , m and a random number ¯ a l i ··· l im ∈ [0 , m -th-order k -dimensional symmetric tensor B , such that all elements of B are in the interval [0 , Z -tensor A = ( a i i ··· i m ) such that a i ··· i m = M if i = · · · = i m = i for all i = 1 , · · · , n, ¯ a l i ··· l im if ( i , · · · , i m ) = σ ( l i , · · · , l im ) with l i , · · · , l im ∈ Γ i , i ∈ [ s − , −B i ··· i m if i , · · · , i m ∈ Γ s , σ ( i , · · · , i m ) denotes all the possible permutation of ( i , · · · , i m ).(iv) Let A h = ( a hi i ··· i m ) be a extended Z -tensor with order 2 m and dimension n such that a hσ ( i i i i ··· i m i m ) = a i i ··· i m , ∀ i , i , · · · , i m ∈ [ n ] , and a hi i ··· i m = 0 otherwise.(v) Let f ( x ) = −A h x m , x ∈ R n . Then solve the SOS programming problem (5.2) by Matlab ToolboxYALMIP [24, 25] and SeDuMi [36].We perform the above procedure to test extended Z-tensors with order m = 3 , , , n = 2500 , , , s = 500 and run one hundred tests. Table 4summarizes the percentage of copositive instances of these extended Z -tensors. Clearly, for fixed order m and dimension n , the percentage of copositive extended Z -tensors increase as the parameter M increase.Moreover, if M is large enough, the generated extended Z -tensor must be positive definite, and so, inparticular is copositive. In this article, we propose an efficient semi-definite program algorithm to compute the maximum H -eigenvalues of even order symmetric W -tensors based on its potential SOS structure. Furthermore, we19able 4: The percentage of copositive instances of randomly generated extended Z -tensors. m = 3 and n = 2500 M
11 12 13 14 15 16Copositivity 12% 35% 68% 91% 96% 99% m = 4 and n = 2000 M
25 28 31 34 37 40Copositivity 2% 11% 36% 63% 89% 99% m = 5 and n = 1500 M
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