ON COMPUTING MAXIMUM/MINIMUM SINGULAR VALUES OF A GENERALIZED TENSOR SUM

Transcription

1 Electronc Transactons on Numercal Analyss. Volume 43, pp , 215. Copyrght c 215, Kent State Unversty. ISSN ETNA Kent State Unversty ON COMPUTING MAXIMUM/MINIMUM SINGULAR VALUES OF A GENERALIZED TENSOR SUM ASUKA OHASHI AND TOMOHIRO SOGABE Abstract. We consder the effcent computaton of the maxmum/mnmum sngular values of a generalzed tensor sum T. The computaton s based on two approaches: frst, the Lanczos bdagonalzaton method s reconstructed over tensor space, whch leads to a memory-effcent algorthm wth a smple mplementaton, and second, a promsng ntal guess gven n Tucer decomposton form s proposed. From the results of numercal experments, we observe that our computaton s useful for matrces beng near symmetrc, and t has the potental of becomng a method of choce for other cases f a sutable core tensor can be gven. Key words. generalzed tensor sum, Lanczos bdagonalzaton method, maxmum and mnmum sngular values AMS subject classfcatons. 65F1 1. Introducton. We consder computng the maxmum/mnmum sngular values of a generalzed tensor sum (1.1) T := I n I m A + I n B I l + C I m I l R lmn lmn, where A R l l, B R m m, C R n n, I m s the m m dentty matrx, the symbol " denotes the Kronecer product, and the matrx T s assumed to be large, sparse, and nonsngular. Such matrces T arse n a fnte-dfference dscretzaton of three-dmensonal constant-coeffcent partal dfferental equatons, such as [ a ( ) + b + c ] u(x, y, z) = g(x, y, z) n Ω, (1.2) u(x, y, z) = on Ω, where Ω = (, 1) (, 1) (, 1), a, b R 3, c R, and the symbol " denotes the elementwse product. If a = (1, 1, 1), then a ( ) =. Wth regard to effcent numercal methods for lnear systems of the form T x = f, see [3, 9]. The Lanczos bdagonalzaton method s wdely nown as an effcent method to compute the maxmum/mnmum sngular values of a large and sparse matrx. For ts recent successful varants, see, e.g., [2, 6, 7, 1, 12], and for other successful methods, see, e.g., [5]. The Lanczos bdagonalzaton method does not requre T and T T themselves but only the results of the matrx-vector multplcatons T v and T T v. Even though one stores, as usual, only the non-zero entres of T, the storage requred grows cubcally wth n under the assumpton that l = m = n and, as t s often the case, that the number of non-zero entres of A, B, and C grows lnearly wth n. In order to avod large memory usage, we consder the Lanczos bdagonalzaton method over tensor space. Advantages of ths approach are a low memory requrement and a very smple mplementaton. In fact, the requred memory for storng T grows only lnearly under the above assumptons. Usng the tensor structure, we present a promsng ntal guess n order to mprove the speed of convergence of the Lanczos bdagonalzaton method over tensor space, whch s a major contrbuton of ths paper. Receved October 31, 213. Accepted July 29, 215. Publshed onlne on October 16, 215. Recommended by H. Sado. Graduate School of Informaton Scence and Technology, Ach Prefectural Unversty, Ibaragabasama, Nagaute-sh, Ach, , Japan (d1412@cs.ach-pu.ac.jp). Graduate School of Engneerng, Nagoya Unversty, Furo-cho, Chusa-u, Nagoya, , Japan (sogabe@na.cse.nagoya-u.ac.jp). 244

2 Kent State Unversty COMPUTING MAX/MIN SINGULAR VALUES OF GENERALIZED TENSOR SUM 245 Ths paper s organzed as follows. In Secton 2, the Lanczos bdagonalzaton method s ntroduced, and an algorthm s presented. In Secton 3, some basc operatons on tensors are descrbed. In Secton 4, we consder the Lanczos bdagonalzaton method over tensor space and propose a promsng ntal guess usng the tensor structure of the matrx T. Numercal experments and concludng remars are gven n Sectons 5 and 6, respectvely. 2. The Lanczos bdagonalzaton method. The Lanczos bdagonalzaton method, whch s due to Golub and Kahan [4], s sutable for computng maxmum/mnmum sngular values. In partcular, the method s wdely used for large and sparse matrces. It employs sequences of projectons of a matrx onto judcously chosen low-dmensonal subspaces and computes the sngular values of the obtaned matrx. By means of the projectons, computng these sngular values s more effcent than for the orgnal matrx snce the obtaned matrx s smaller and has a smpler structure. For a matrx M R l n (l n), the Lanczos bdagonalzaton method dsplayed n Algorthm 1 calculates a sequence of vectors p R n and q R l and scalars α and β, where = 1, 2,...,. Here, represents the number of bdagonalzaton steps and s typcally much smaller than ether one of the matrx dmensons l and n. Algorthm 1 The Lanczos bdagonalzaton method [4]. 1: Choose an ntal vector p 1 R n such that p 1 2 = 1. 2: q 1 := Mp 1 ; 3: α 1 := q 1 2 ; 4: q 1 := q 1 /α 1 ; 5: for = 1, 2,..., do 6: r := M T q α p ; 7: β := r 2 ; 8: p +1 := r /β ; 9: q +1 := Mp +1 β q ; 1: α +1 := q +1 2 ; 11: q +1 := q +1 /α +1 ; 12: end for (2.1) After steps, Algorthm 1 yelds the followng decompostons: MP = Q D, M T Q = P D T + r e T, where the vectors e and r denote the -th canoncal and the -th resdual vector n Algorthm 1, respectvely, and the matrces D, P, and Q are gven as α 1 β 1 α 2 β 2 D = (2.2) R, α 1 β 1 α P = (p 1, p 2,..., p ) R n, Q = (q 1, q 2,..., q ) R l. Here, P and Q are column orthogonal matrces,.e., P TP = Q T Q = I. Now, the sngular trplets of the matrces M and D are explaned. Let σ (M) 1, σ (M) 2,...,σ n (M) be the sngular values of M such that σ (M) 1 σ (M) 2 σ n (M). Moreover, let u (M) R l

3 Kent State Unversty 246 A. OHASHI AND T. SOGABE and v (M) R n, where = 1, 2,..., n, be the left and rght sngular vectors correspondng to, v (M) } s referred to as a sngular the sngular values σ (M), respectvely. Then, {σ (M), u (M) trplet of M, and the relatons between M and ts sngular trplets are gven as Mv (M) = σ (M) u (M), M T u (M) = σ (M) v (M), where = 1, 2,..., n. Smlarly, wth regard to D n (2.2), let {σ (D ), u (D ), v (D ) } be the sngular trplets of D. Then, the relatons between D and ts sngular trplets are (2.3) D v (D ) = σ (D ) u (D ), D T u (D ) = σ (D ) v (D ), where = 1, 2,...,. Moreover, { σ (M), ũ (M), ṽ (M) } denotes the approxmate sngular trplet of M. They are determned from the sngular trplet of D as follows: (2.4) σ (M) := σ (D ), ũ (M) := Q u (D ), ṽ (M) := P v (D ). Then, t follows from (2.1), (2.3), and (2.4) that (2.5) Mṽ (M) M T ũ (M) = σ (M) ũ (M), = σ (M) ṽ (M) + r e T u (D ), where = 1, 2,...,. Equatons (2.5) mply that the approxmate sngular trplet { σ (M), ũ (M), ṽ (M) } s acceptable for the sngular trplet {σ (M), u (M), v (M) } f the value of r 2 e T u(d ) s suffcently small. 3. Some basc operatons on tensors. Ths secton provdes a bref explanaton of tensors. For further detals, see, e.g., [1, 8, 11]. A tensor s a multdmensonal array. In partcular, a frst-order tensor s a vector, a secondorder tensor s a matrx, and a thrd-order tensor, whch s manly used n ths paper, has three ndces. Thrd-order tensors are denoted by X, Y, P, Q, R, and S. An element (, j, ) of a thrd-order tensor X s denoted by x j. When the sze of a tensor X s I J K, the ranges of, j, and are = 1, 2,..., I, j = 1, 2,..., J, and = 1, 2,..., K, respectvely. We descrbe the defntons of some basc operatons on tensors. Let x j and y j be elements of the tensors X, Y R I J K. Then, addton s defned by elementwse summaton of X and Y: (X + Y) j := x j + y j, scalar tensor multplcaton s defned by the product of the scalar λ and each element of X : (λx ) j := λx j, and the dot product s defned by the summaton of elementwse products of X and Y: (X, Y) := I J =1 j=1 =1 K (X Y) j, where the symbol " denotes the elementwse product. Then, the norm s defned as X := (X, X ).

4 Kent State Unversty COMPUTING MAX/MIN SINGULAR VALUES OF GENERALIZED TENSOR SUM 247 Let us defne some tensor multplcatons: an n-mode product, whch s denoted by the symbol n ", s a products of a matrx M and a tensor X. The n-mode product for a thrd-order tensor has three dfferent types. The 1-mode product of X R I J K and M R P I s defned as (X 1 M) pj := I x j m p, the 2-mode product of X R I J K and M R P J s defned as (X 2 M) p := =1 J x j m pj, and the 3-mode product of X R I J K and M R P K s defned as (X 3 M) jp := j=1 K x j m p, where = 1, 2,..., I, j = 1, 2,..., J, = 1, 2,..., K, and p = 1, 2,..., P. Fnally, the operator vec vectorzes a tensor by combnng all column vectors of the tensor nto one long vector: =1 vec : R I J K R IJK, and the operator vec 1 reshapes a tensor from one long vector: vec 1 : R IJK R I J K. We wll see that the vec 1 -operator plays an mportant role n reconstructng the Lanczos bdagonalzaton method over tensor space. 4. The Lanczos bdagonalzaton method over tensor space wth a promsng ntal guess The Lanczos bdagonalzaton method over tensor space. If the Lanczos bdagonalzaton method s appled to the generalzed tensor sum T n (1.1), the followng matrx-vector multplcatons are requred: (4.1) T p = (I n I m A + I n B I l + C I m I l ) p, T T p = ( I n I m A T + I n B T I l + C T I m I l ) p, where p R lmn. In an mplementaton, however, computng the multplcatons (4.1) becomes complcated snce t requres the non-zero structure of a large matrx T. Here, the relatons between the vec-operator and the Kronecer product are represented by (I n I m A) vec(p) = vec(p 1 A), (I n B I l ) vec(p) = vec(p 2 B), (C I m I l ) vec(p) = vec(p 3 C), where P R l m n s such that vec(p) = p. Usng these relatons, the multplcatons (4.1) can be descrbed by (4.2) T p = vec (P 1 A + P 2 B + P 3 C), T T p = vec ( P 1 A T + P 2 B T + P 3 C T).

5 Kent State Unversty 248 A. OHASHI AND T. SOGABE Then, an mplementaton based on (4.2) only requres the non-zero structures of the matrces A, B, and C beng much smaller than of T, and thus t s smplfed. We now consder the Lanczos bdagonalzaton method over tensor space. Applyng the vec 1 -operator to (4.2) yelds vec 1 (T p) = P 1 A + P 2 B + P 3 C, vec 1 ( T T p ) = P 1 A T + P 2 B T + P 3 C T. Then, the Lanczos bdagonalzaton method over tensor space for T s obtaned and summarzed n Algorthm 2. Algorthm 2 The Lanczos bdagonalzaton method over tensor space. 1: Choose an ntal tensor P 1 R l m n such that P 1 = 1. 2: Q 1 := P 1 1 A + P 1 2 B + P 1 3 C; 3: α 1 := Q 1 ; 4: Q 1 := Q 1 /α 1 ; 5: for = 1, 2,..., do 6: R := Q 1 A T + Q 2 B T + Q 3 C T α P ; 7: β := R ; 8: P +1 := R /β ; 9: Q +1 := P +1 1 A + P +1 2 B + P +1 3 C β Q ; 1: α +1 := Q +1 ; 11: Q +1 := Q +1 /α +1 ; 12: end for The maxmum/mnmum sngular values of T are computed by a sngular value decomposton of the matrx D n (2.2), whose entres α and β are obtaned from Algorthm 2. The convergence of Algorthm 2 can be montored by R e T u (D ) (4.3), where u (D ) s computed by the sngular value decomposton for D n (2.2) A promsng ntal guess. We consder utlzng the egenvectors of the small matrces A, B, and C for determnng a promsng ntal guess for Algorthm 2. We propose the ntal guess P 1 that s gven n Tucer decomposton form: (4.4) P 1 := S 1 P A 2 P B 3 P C, where S R s the core tensor such that =1 j=1 =1 s j = 1 and s j, P A = [x (A) M, x (A) m ], P B = [x (B), x (B) jm ], and P C = [x (C), x (C) ]. The rest of ths secton M m defnes the vectors x (A) M, x (B), x (C), x (A) M m, x(b), and x(c) m. Let {λ (A), x (A) }, {λ (B) j, x (B) j jm }, and {λ (C), x (C) and C, respectvely. Then, x (A) M, x (B), and x (C) M egenvalues λ (A) M, λ (B), and λ (C) M of A, B, and C, where { λ (A) ( M,, M ) = argmax + λ (B) j (,j,) } be egenpars of the matrces A, B, are the egenvectors correspondng to the + λ (C) }.

6 Kent State Unversty Smlarly, x (A) λ (B) jm, and λ(c) m COMPUTING MAX/MIN SINGULAR VALUES OF GENERALIZED TENSOR SUM 249, and x(c) m of A, B, and C, where m, x(b) jm are the egenvectors correspondng to the egenvalues λ(a) ( m, j m, m ) = argmn (,j,) { λ (A) + λ (B) j + λ (C) Here, we note that the egenvector correspondng to the maxmum absolute egenvalue of T s gven by x (C) x (B) M x (A) M and that the egenvector correspondng to the mnmum absolute egenvalue of T s gven by x (C) m x(b) jm x(a) m. 5. Numercal examples. In ths secton, we report the results of numercal experments usng test matrces gven below. All computatons were carred out usng MATLAB verson R211b on a HP Z8 worstaton wth two 2.66 GHz Xeon processors and 24GB of memory runnng under a Wndows 7 operatng system. The maxmum/mnmum sngular values were computed by Algorthm 2 wth random ntal guesses and wth the proposed ntal guess (4.4). From (4.3), the stoppng crtera we used were R e T u(d ) M < 1 1 for the maxmum sngular value σ (D ) M and R e T u(d ) m < 1 1 for the mnmum sngular value σ (D ) m. Algorthm 2 was stopped when both crtera were satsfed Test matrces. The test matrces T arse from a 7-pont central dfference dscretzaton of the PDE (1.2) over an (n + 1) (n + 1) (n + 1) grd, and they are wrtten as a generalzed tensor sum of the form }. T = I n I n A + I n B I n + C I n I n R n3 n 3, where A, B, C R n n. To be specfc, the matrces A, B, and C are gven by (5.1) (5.2) A = 1 h 2 a 1M h b 1M 2 + ci n, B = 1 h 2 a 2M h b 2M 2, (5.3) C = 1 h 2 a 3M h b 3M 2, where a and b ( = 1, 2, 3) correspond to the -th elements of a and b n (1.2), respectvely, and h, M 1, and M 2 are gven as (5.4) (5.5) h = 1 n + 1, M 1 = R n n, M 2 = R n n As can be seen from (5.1) (5.5), the matrx T has hgh symmetry when a 2 s much larger than b 2 and low symmetry else. m,

7 Kent State Unversty 25 A. OHASHI AND T. SOGABE 5.2. Intal guesses used n the numercal examples. In our numercal experments, we set S n (4.4) to be a dagonal tensor,.e., s j = except for = j = = 1 and = j = = 2. Then, the proposed ntal guess (4.4) s represented by the followng convex combnaton of ran-one tensors: (5.6) ( ) ( P 1 = s 111 x (A) M x (B) x (C) + s 222 x (A) M m x(b) jm ) x(c), m where the symbol " denotes the outer product and s s 222 = 1 wth s 111, s 222. As seen n Secton 4.2, the vectors x (A) M, x (B), x (C), x (A) M m, x(b), and x(c) are determned jm m by specfc egenvectors of the matrces A, B, and C. Snce these matrces are trdagonal Toepltz matrces, t s wdely nown that the egenvalues and egenvectors are gven n analytcal form as follows: let D be a trdagonal Toepltz matrx d 1 d 3 d 2 d 1 d 3 D = R n n, d 2 d 1 d 3 d 2 d 1 where d 2 and d 3. Then, the egenvalues λ (D) λ (D) ( π = d 1 + 2dcos n + 1 are computed by ), = 1, 2,..., n, where d = sgn(d 2 ) d 2 d 3, and the correspondng egenvectors x (D) ( ) ( ) d3 2 π sn n + 1 x (D) 2 = n + 1 d ( 2 ) 1 d3 ( d3 d 2 d 2 ) n sn ( 2π n + 1. sn ) ( ) nπ n + 1 are gven by, = 1, 2,..., n Numercal results. Frst, we use the matrx T wth the parameters a = (1, 1, 1), b = (1, 1, 1), and c = 1 n (5.1) (5.3). The convergence hstory of Algorthm 2 wth the proposed ntal guess (5.6) usng s 111 = s 222 =.5 s dsplayed n Fgure 5.1 wth the number of teratons requred by Algorthm 2 on the horzontal axs and the log 1 of the resdual norms on the vertcal axs. Here, the -th resdual norms are computed by R e T u(d) M for the maxmum sngular value and R e T u(d) m for the mnmum sngular value, and the sze of the tensor n Algorthm 2 s As llustrated n Fgure 5.1, Algorthm 2 wth the proposed ntal guess requred 68 teratons for the maxmum sngular value and 46 teratons for the mnmum sngular value. From Fgure 5.1, we observe a smooth convergence behavor and faster convergence for the maxmum sngular value. Next, usng the matrx T wth the same parameters as for Fgure 5.1, the varaton n the number of teratons s dsplayed n Fgure 5.2, where the dependence of the number

8 Kent State Unversty COMPUTING MAX/MIN SINGULAR VALUES OF GENERALIZED TENSOR SUM 251 log 1 of the resdual norm Convergence to maxmum sngular value Convergence to mnmum sngular value Number of teratons FIG The convergence hstory for Algorthm 2 wth the proposed ntal guess. of teratons on the value s 111 n the proposed ntal guess s gven n Fgure 5.2(a), and the dependence on the tensor sze s gven n Fgure 5.2(b). For comparson, the numbers of teratons requred by Algorthm 2 wth ntal guesses beng random numbers are also dsplayed n Fgure 5.2(b). In Fgure 5.2(a), the horzontal axs denotes the value of s 111, varyng from to 1 ncrementally wth a stepsze of.1, and the vertcal axs denotes the number of teratons requred by Algorthm 2 wth the proposed ntal guess. Here, the value of s 222 s computed by s 222 = 1 s 111, and the matrx T used for Fgure 5.2(a) s obtaned from the dscretzaton of the PDE (1.2) over a grd. On the other hand, n Fgure 5.2(b), the horzontal axs denotes the value of n, where the sze of the tensor s n n n (n = 5, 1,..., 35), and the vertcal axs denotes the number of teratons requred by Algorthm 2 wth the proposed ntal guess usng s 111 = s 222 =.5. Here, ths ntal guess s referred to as the typcal case. Number of teratons Convergence to maxmum sngular value Convergence to mnmum sngular value The value of s 111 (a) Number of teraton versus the value of s 111. Number of teratons Typcal case(s 111 =.5) Random numbers 1 Random numbers The value of n (sze of tensor: n n n) (b) Number of teraton versus the tensor sze. FIG The varaton n the number of teratons for the matrx T wth a = b = (1, 1, 1), c = 1. In Fgure 5.2(a), the number of teratons hardly depends on the value of s 111, but there s a bg dfference n ths number between the cases s 111 =.9 and s 111 = 1. We therefore ran Algorthm 2 wth the proposed ntal guess usng the values s 111 =.9,.91,...,.99. As a result, we confrm that the numbers of teratons are almost the same as those for the cases s 111 =,.1,...,.9. From these results, we fnd almost no dependency of the number of teratons on the choce of the parameter s 111, and ths mples the robustness of the proposed ntal guess. It seems that the hgh number of teratons requred for the case

9 Kent State Unversty 252 A. OHASHI AND T. SOGABE s 111 = 1 s due to that the gven ntal guess only has a very small component of the sngular vector correspondng to the mnmum sngular value. In fact, for a symmetrc matrx, such a choce means that the proposed ntal guess ncludes no component of the sngular vector correspondng to the mnmum sngular value. In Fgure 5.2(b) we observe that Algorthm 2 n the typcal case requres fewer teratons than wth an ntal guess of random numbers, and the gap grows as n ncreases. In what follows, we use matrces T wth hgher or lower symmetry than for the matrces used n Fgures 5.1 and 5.2. A matrx T wth hgher symmetry s created from the parameters a = (1, 1, 1), b = (1, 1, 1), and c = 1, and a matrx T wth lower symmetry from the parameters a = (1, 1, 1), b = (1, 1, 1), and c = 1. The varaton n the number of teratons by the value of s 111 n the proposed ntal guess s presented n Fgure 5.3. Here, the matrces T arse from the dscretzaton of the PDE (1.2) wth the above parameters over a grd. 8 7 Convergence to maxmum sngular value Convergence to mnmum sngular value 8 7 Convergence to maxmum sngular value Convergence to mnmum sngular value Number of teratons Number of teratons The value of s The value of s 111 (a) T wth hgher symmetry. (b) T wth lower symmetry. FIG The varaton n the number of teratons versus the value of s 111. In Fgure 5.3, the varaton n the number of teratons for the matrces wth hgh and low symmetry showed smlar tendences as n Fgure 5.2(a). Furthermore, the varaton n the number of teratons requred by Algorthm 2 wth the proposed ntal guess usng the values s 111 =.9,.91,...,.99 n Fgure 5.3 has the same behavor as that n Fgure 5.2(a). For the low-symmetry case, the choce s 111 = 1 was optmal unle the other cases. In the followng example, the varaton n the number of teratons versus the tensor sze s dsplayed n Fgure 5.4. For comparson, we ran Algorthm 2 wth several ntal guesses: random numbers and the proposed one wth the typcal case s 111 =.5. Accordng to Fgure 5.4(a), Algorthm 2 n the typcal case requred fewer teratons than for the ntal guesses usng random numbers when T had hgher symmetry. On the other hand, Fgure 5.4(b) ndcates that Algorthm 2 n the typcal case requred as many teratons as for a random ntal guess when T had lower symmetry. From Fgures 5.2(b) and 5.4, we observe that the ntal guess usng s 111 < 1 mproves the speed of convergence of Algorthm 2 except for the case where T has lower symmetry. As can be observed n Fgure 5.4(b), the typcal case shows no advantage over the random ntal guess for the low-symmetry matrx. On the other hand, for some cases the proposed ntal guess could stll become a method of choce, for nstance, for the case s 111 = 1 dsplayed n Fgure 5.5. It s lely, though t requrng further nvestgaton, that the result n Fgure 5.5 ndcates a potental for mprovement of the proposed ntal guess (4.4) even for the low-symmetry case.

10 Kent State Unversty COMPUTING MAX/MIN SINGULAR VALUES OF GENERALIZED TENSOR SUM 253 Number of teratons Typcal case(s 111 =.5) Random numbers 1 Random numbers 2 Number of teratons Typcal case(s 111 =.5) Random numbers 1 Random numbers The value of n (sze of tensor: n n n) (a) T wth hgher symmetry The value of n (sze of tensor: n n n) (b) T wth lower symmetry. FIG The varaton n the number of teratons versus the tensor sze. Number of teratons Typcal case(s 111 =.5) Random numbers 1 Random numbers 2 Sutable case(s 111 =1) The value of n (sze of tensor: n n n) FIG The varaton n the number of teratons requred by Algorthm 2 n the sutable case versus the tensor sze when T has lower symmetry. In fact, we only used a dagonal tensor as ntal guess, whch s a subtensor of the core tensor. For low-symmetry matrces, an expermental nvestgaton of the optmal choce of a full core tensor wll be consdered n future wor. 6. Concludng remars. In ths paper, frst, we derved the Lanczos bdagonalzaton method over tensor space from the conventonal Lanczos bdagonalzaton method usng the vec 1 -operator n order to compute the maxmum/mnmum sngular values of a generalzed tensor sum T. The resultng method acheved a low memory requrement and a very smple mplementaton snce t only requred the non-zero structure of the matrces A, B, and C. Next, we proposed an ntal guess gven n Tucer decomposton form usng egenvectors correspondng to the maxmum/mnmum egenvalues of T. Computng the egenvectors of T was easy snce the egenpars of T were obtaned from the egenpars of A, B, and C. Fnally, from the results of the numercal experments, we showed that the maxmum/mnmum sngular values of T were successfully computed by the Lanczos bdagonalzaton method over tensor space wth some of the proposed ntal guesses. We see that the proposed ntal guesses mproved the speed of convergence of the Lanczos bdagonalzaton method over tensor space for the hgh-symmetry case and that t could become a method of choce for other cases f a sutable core tensor can be found. Future wor s devoted to an expermental nvestgatons usng the full core tensor n the proposed ntal guess n order to choose an optmal ntal guess for low-symmetry matrces. If the generalzed tensor sum (1.1) s suffcently close to a symmetrc matrx, our ntal guess

11 Kent State Unversty 254 A. OHASHI AND T. SOGABE wors very well, but n general, restartng technques are mportant for a further mprovement of the speed of convergence to the mnmum sngular value. In ths case, restartng technques should be combned not n vector- but n tensor space. Thus, constructng a general framewor n tensor space and combnng Algorthm 2 wth successful restartng technques, e.g., [2, 6, 7] are topcs of future wor. Wth regards to other methods, the presented approach may be appled to other successful varants of the Lanczos bdagonalzaton method, e.g., [1] and to Jacob-Davdson-type sngular value decomposton methods, e.g., [5]. Acnowledgments. Ths wor has been supported n part by JSPS KAKENHI (Grant No ). We wsh to express our grattude to Dr. D. Savostyanov, Unversty of Southampton, for constructve comments at the NASCA213 conference. We are grateful to Dr. T. S. Usuda and Dr. H. Yoshoa of Ach Prefectural Unversty for ther support and encouragement. We would le to than the anonymous referees for nformng us of reference [11] and many useful comments that enhanced the qualty of the manuscrpt. REFERENCES [1] B. W. BADER AND T. G. KOLDA, Algorthm 862: MATLAB tensor classes for fast algorthm prototypng, ACM Trans. Math. Software, 32 (26), pp [2] J. BAGLAMA AND L. REICHEL, An mplctly restarted bloc Lanczos bdagonalzaton method usng Leja shfts, BIT Numer. Math., 53 (213), pp [3] J. BALLANI AND L. GRASEDYCK, A projecton method to solve lnear systems n tensor format, Numer. Lnear Algebra Appl., 2 (213), pp [4] G. GOLUB AND W. KAHAN, Calculatng the sngular values and pseudo-nverse of a matrx, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), pp [5] M. E. HOCHSTENBACH, A Jacob-Davdson type method for the generalzed sngular value problem, Lnear Algebra Appl., 431 (29), pp [6] Z. JIA AND D. NIU, A refned harmonc Lanczos bdagonalzaton method and an mplctly restarted algorthm for computng the smallest sngular trplets of large matrces, SIAM J. Sc. Comput., 32 (21), pp [7] E. KOKIOPOULOU, C. BEKAS, AND E. GALLOPOULOS, Computng smallest sngular trplets wth mplctly restarted Lanczos bdagonalzaton, Appl. Numer. Math., 49 (24), pp [8] T. G. KOLDA AND B. W. BADER, Tensor decompostons and applcatons, SIAM Rev., 51 (29), pp [9] D. KRESSNER AND C. TOBLER, Krylov subspace methods for lnear systems wth tensor product structure, SIAM J. Matrx Anal. Appl., 31 (21), pp [1] D. NIU AND X. YUAN, A harmonc Lanczos bdagonalzaton method for computng nteror sngular trplets of large matrces, Appl. Math. Comput., 218 (212), pp [11] B. SAVAS AND L. ELDEN, Krylov-type methods for tensor computatons I, Lnear Algebra Appl., 438 (213), pp [12] M. STOLL, A Krylov-Schur approach to the truncated SVD, Lnear Algebra Appl., 436 (212), pp