Tensor Low-Rank Completion and Invariance of the Tucker Core

Transcription

1 Tensor Low-Rank Completion and Invariance of the Tucker Core Shuzhong Zhang Department of Industrial & Systems Engineering University of Minnesota Joint work with Bo JIANG, Shiqian MA, and Fan YANG IMA Annual Program Year Workshop Optimization and Parsimonious Modeling January 28, 2016 Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

2 Part I. Matricization and Tensor Completion Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

3 Tensor CP-Rank Consider d-th order tensor F C n 1 n 2 n d, its CP-rank (denoted as rank CP (F )) is the smallest integer r exhibiting the following decomposition F = r a 1,i a 2,i a d,i i=1 where a k,i C n i for k = 1,..., d and i = 1,..., r. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

4 Tensor CP-Rank Consider d-th order tensor F C n 1 n 2 n d, its CP-rank (denoted as rank CP (F )) is the smallest integer r exhibiting the following decomposition F = r a 1,i a 2,i a d,i i=1 where a k,i C n i for k = 1,..., d and i = 1,..., r. If F is in the real domain, then its real CP-rank rankcp R (F ) is the above decomposition where all the vectors are real. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

5 Tensor CP-Rank Consider d-th order tensor F C n 1 n 2 n d, its CP-rank (denoted as rank CP (F )) is the smallest integer r exhibiting the following decomposition F = r a 1,i a 2,i a d,i i=1 where a k,i C n i for k = 1,..., d and i = 1,..., r. If F is in the real domain, then its real CP-rank rankcp R (F ) is the above decomposition where all the vectors are real. It is well known that in general when d 3. rank CP (F ) < rank R CP (F ) Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

6 Low-CP-Rank Tensor Recovery Tensor Completion min s.t. rank CP (X) X C n 1 n 2 n d L(X) = b. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

7 Low-CP-Rank Tensor Recovery Tensor Completion min s.t. rank CP (X) X C n 1 n 2 n d L(X) = b. Robust Tensor Recovery min rank CP (Y) + λ Z 0 s.t. Y, Z C n 1 n 2 n d Y + Z = F. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

8 Hardness of the CP-Rank Computation Consider the following tensor (Kruskal, 1989): and all other components are zero. (1,1,1): 1 (4,2,1): 1 (7,3,1): 1 (1,4,2): 1 (4,5,2): 1 (7,6,2): 1 (1,7,3): 1 (4,8,3): 1 (7,9,3): 1 (2,1,4): 1 (5,2,4): 1 (8,3,4): 1 (2,4,5): 1 (5,5,5): 1 (8,6,5): 1 (2,7,6): 1 (5,8,6): 1 (8,9,6): 1 (3,1,7): 1 (6,2,7): 1 (9,3,7): 1 (3,4,8): 1 (6,5,8): 1 (9,6,8): 1 (3,7,9): 1 (6,8,9): 1 (9,9,9): 1 The CP-rank of the above tensor is only known to be between 19 and 23. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

9 Matricized Ranks For F C n 1 n 2 n d, its mode- j matricization F( j) is obtained by arranging the j-th index of F as the column index of F( j) and merging other indices of F as the row index of F( j). The Tucker rank of F is defined as the vector (rank(f(1)),..., rank(f(d))). The averaged Tucker rank rank n (F ) := 1 d d rank(f( j)) j=1 Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

10 The Low-n-Rank Recovery Liu, Musialski, Wonka, Ye, 2009; Gandy, Recht, Yamada, 2011: min s.t. rank CP (X) X C n 1 n 2 n d L(X) = b = Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

11 The Low-n-Rank Recovery Liu, Musialski, Wonka, Ye, 2009; Gandy, Recht, Yamada, 2011: min s.t. rank CP (X) X C n 1 n 2 n d L(X) = b = min s.t. rank n (X) X C n 1 n 2 n d L(X) = b Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

12 The Low-n-Rank Recovery Liu, Musialski, Wonka, Ye, 2009; Gandy, Recht, Yamada, 2011: min s.t. rank CP (X) X C n 1 n 2 n d L(X) = b = min s.t. rank n (X) X C n 1 n 2 n d L(X) = b and its convexified version min 1 dj=1 d X( j) s.t. X C n 1 n 2 n d L(X) = b. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

13 The Low-n-Rank Recovery Liu, Musialski, Wonka, Ye, 2009; Gandy, Recht, Yamada, 2011: min s.t. rank CP (X) X C n 1 n 2 n d L(X) = b = min s.t. rank n (X) X C n 1 n 2 n d L(X) = b and its convexified version min 1 dj=1 d X( j) s.t. X C n 1 n 2 n d L(X) = b. We have rank n (X) rank CP (X), but no reasonable upper bound is known. The method works well when rank CP (X) is relatively small. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

14 Fair and Square We may also group the indices by equal parts and then unfold it into a matrix. Let us call it Square Unfolding (Mu, Huang, Wright, Goldfarb, 2013). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

15 Fair and Square We may also group the indices by equal parts and then unfold it into a matrix. Let us call it Square Unfolding (Mu, Huang, Wright, Goldfarb, 2013). Tomioka, Suzuki, Hayashi, 2011: It takes O(rn d 1 ) observations for the low-n-rank model to complete the tensor. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

16 Fair and Square We may also group the indices by equal parts and then unfold it into a matrix. Let us call it Square Unfolding (Mu, Huang, Wright, Goldfarb, 2013). Tomioka, Suzuki, Hayashi, 2011: It takes O(rn d 1 ) observations for the low-n-rank model to complete the tensor. Mu, Huang, Wright, Goldfarb, 2013: It takes O(r d 2 n d 2 ) observations for the square unfolding model to complete the tensor. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

17 The M-Rank Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

18 The M-Rank Definition For an even-order tensor F C n 1 n 2 n 2d, the M-decomposition is to find some tensors A i C n 1 n d, B i C n d+1 n 2d with i = 1,..., r such that F = r A i B i. i=1 Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

19 The M-Rank Definition For an even-order tensor F C n 1 n 2 n 2d, the M-decomposition is to find some tensors A i C n 1 n d, B i C n d+1 n 2d with i = 1,..., r such that F = r A i B i. i=1 Definition Given an even-order tensor F C n 1 n 2 n 2d, its M -rank is the smallest rank of all square unfolding matrices, i.e., rank M (F ) = min rank (M(F π)), π Π(1,...,2d) and the M + -rank is the largest rank of all square unfolding matrices: rank M +(F ) = max rank (M(F π)). π Π(1,...,2d) Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

20 The Symmetric CP-Rank Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

21 The Symmetric CP-Rank Definition Given a d-th order n-dimensional super-symmetric complex-valued tensor F, its symmetric CP-rank (denoted by rank S CP (F )) is the smallest integer r such that with a i C n, i = 1,..., r. F = r i=1 a i a i } {{ } d Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

22 The Symmetric CP-Rank Definition Given a d-th order n-dimensional super-symmetric complex-valued tensor F, its symmetric CP-rank (denoted by rank S CP (F )) is the smallest integer r such that with a i C n, i = 1,..., r. F = r i=1 Obviously, rank CP (F ) rank S CP (F ). a i a i } {{ } d Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

23 The Symmetric CP-Rank Definition Given a d-th order n-dimensional super-symmetric complex-valued tensor F, its symmetric CP-rank (denoted by rank S CP (F )) is the smallest integer r such that with a i C n, i = 1,..., r. F = r i=1 Obviously, rank CP (F ) rank S CP (F ). a i a i } {{ } d Open Problem: rank CP (F ) = rank S CP (F )? Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

24 The Symmetric CP-Rank Definition Given a d-th order n-dimensional super-symmetric complex-valued tensor F, its symmetric CP-rank (denoted by rank S CP (F )) is the smallest integer r such that with a i C n, i = 1,..., r. F = r i=1 Obviously, rank CP (F ) rank S CP (F ). a i a i } {{ } d Open Problem: rank CP (F ) = rank S CP (F )? Zhang, Huang, Qi, 2014: rank CP (F ) = rank S CP (F ) if rank CP (F ) d. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

25 The Symmetric M-Rank ab + ba = 1 2 (a + b)(a + b) 1 2 (a b)(a b). Definition For an even-order super-symmetric tensor F S n2d, its symmetric M-rank is the smallest r such that F = r B i B i, B i C nd, i = 1,..., r. i=1 The strongly symmetric M-rank is the smallest integer r such that F = r A i A i, A i S nd, i = 1,..., r. i=1 Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

26 Theorem (Jiang, Ma, and Z.; 2015) For an even-order super-symmetric tensor F S n2d, we have rank M (F ) = rank S M (F ) = rank S S M (F ). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

27 Theorem (Jiang, Ma, and Z.; 2015) For an even-order super-symmetric tensor F S n2d, we have rank M (F ) = rank S M (F ) = rank S S M (F ). Theorem (Jiang, Ma, and Z.; 2015) Suppose F S n2d. We have rank M (F ) = 1 rank S CP (F ) = 1. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

28 Theorem (Jiang, Ma, and Z.; 2015) For an even-order super-symmetric tensor F S n2d, we have rank M (F ) = rank S M (F ) = rank S S M (F ). Theorem (Jiang, Ma, and Z.; 2015) Suppose F S n2d. We have rank M (F ) = 1 rank S CP (F ) = 1. Theorem (Jiang, Ma, and Z.; 2015) For any given F S n4, it holds that rank M (F ) rank S CP (F ) (n + 4n 2 ) rank M (F ). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

29 Non-Symmetric Tensors Theorem (Jiang, Ma, and Z.; 2015) Suppose F C n 1 n 2 n 3 n 4 with n 1 n 2 n 3 n 4. Then for any permutation π of (1, 2, 3, 4) it holds that rank(m(f π )) rank CP (F π ) n 1 n 3 rank(m(f π )). Moreover, the inequalities above can be sharpened to rank M +(F ) rank CP (F ) n 1 n 3 rank M (F ). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

30 Non-Symmetric Tensors Theorem (Jiang, Ma, and Z.; 2015) Suppose F C n 1 n 2 n 3 n 4 with n 1 n 2 n 3 n 4. Then for any permutation π of (1, 2, 3, 4) it holds that rank(m(f π )) rank CP (F π ) n 1 n 3 rank(m(f π )). Moreover, the inequalities above can be sharpened to rank M +(F ) rank CP (F ) n 1 n 3 rank M (F ). As a consequence, rank CP (F π ) rankcp R (F π) n 1 n 3 rank R (M(F π )) = n 1 n 3 rank(m(f π )) n 1 n 3 rank CP (F π ). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

31 Tightness of the Bounds Let us consider a fourth order tensor F C n 1 n 2 n 3 n 4 such that F = A B for some matrices A C n 1 n 2 and B C n 3 n 4. Denote r 1 = rank(a), r 2 = rank(b). Then, the following holds: (i) The Tucker rank of F is (r 1, r 1, r 2, r 2 ); (ii) rank M +(F ) = r 1 r 2 and rank M (F ) = 1; (iii) rank CP (F ) = r 1 r 2. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

32 Higher Dimensions In general, for an even order tensor F C n 1 n 2 n 2d with F = A 1 A 2 A d for some matrices A i C n 2i 1 n 2i. Denoting r i = rank(a i ) for i = 1,..., d, we have rank CP (F ) = rank M +(F ) = r 1 r 2 r d. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

33 Part II. Numerical Experiments Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

34 How does the M-Rank relate to the CP-Rank? We generate random tensors r T = a i b i c i d i C n 1 n 2 n 3 n 4. i=1 Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

35 How does the M-Rank relate to the CP-Rank? We generate random tensors r T = a i b i c i d i C n 1 n 2 n 3 n 4. i=1 For each size, we generate 20 instances and compute the average ranks: CP-rank Tucker rank M + -rank M -rank Dimension r = 18, CP-rank 18 (15,15,15,15) Dimension r = 18, CP-rank 18 (15,15,18,18) Dimension r = 30, CP-rank 30 (20,20,20,20) Dimension r = 30, CP-rank 30 (20,20,25,25) Dimension r = 40, CP-rank 40 (25,25,30,30) Dimension r = 40, CP-rank 40 (30,30,30,30) Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

36 Different Type of Random Tensors Next, we randomly generate tensors of the form r T = A i B i, with rank(a i ) = rank(b i ) = k, i = 1,..., r, and we do the same tests: i=1 Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

37 Different Type of Random Tensors Next, we randomly generate tensors of the form r T = A i B i, with rank(a i ) = rank(b i ) = k, i = 1,..., r, and we do the same tests: i=1 CP-rank Tucker rank M + -rank M -rank Dimension r = k = 2, CP-rank 8 (4,4,4,4) 8 2 r = k = 3, CP-rank 27 (9,9,9,9) 27 3 r = k = 4, CP-rank 64 (16,16,16,16) 64 4 Dimension r = k = 3, CP-rank 27 (9,9,9,9) 27 3 r = k = 4, CP-rank 64 (15,15,16,16) 64 4 r = k = 5, CP-rank 125 (20,20,20,20) Dimension r = k = 3, CP-rank 27 (9,9,9,9) 27 3 r = k = 4, CP-rank 64 (16,16,16,16) 64 4 r = k = 5, CP-rank 125 (20,20,25,25) Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

38 Low CP-Rank Recovery Recall the low CP-rank recovery problem min rank CP (X) s.t. L(X) = b. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

39 Low CP-Rank Recovery Recall the low CP-rank recovery problem min rank CP (X) s.t. L(X) = b. The alternative convexifications min 1 dj=1 d X( j) s.t. L(X) = b min s.t. M(X) L(X) = b low-n-rank completion low-m-rank completion Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

40 X 0 = r i=1 a i b i c i d i C n 1 n 2 n 3 n 4 ; RelErr := X X 0 F X 0 F. sampling ratio low-n-rank completion low-m-rank completion Rel Err Tucker rank Rel Err M + -rank M -rank Dimension , r = 4, CP-rank 4 70% 7.26e-006 (4, 4, 4, 4) 3.49e % 1.45e-005 (4, 4, 4, 4) 6.76e % 1.96e-005 (4, 4, 4, 4) 2.37e Dimension , r = 8, CP-rank 8 70% 1.67e-005 (8, 8, 8, 8) 6.60e % 1.34e-001 (20, 20, 20, 20) 1.59e % 5.87e-001 (20, 20, 20, 20) 4.24e Dimension , r = 12, CP-rank 12 70% 1.45e-001 (20, 20, 20, 20) 8.58e % 4.20e-001 (20, 20, 20, 20) 3.46e % 6.99e-001 (20, 20, 20, 20) 6.75e Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

41 X 0 = r i=1 A i B i ; RelErr := X X 0 F X 0 F ; samply ratio = 30%. rank CP (X 0 ) RelErr CPU rank M +(X ) rank M (X ) Dimension r = 12, rank CP (X 0 ) e Dimension r = 15, rank CP (X 0 ) e Dimension r = 24, rank CP (X 0 ) e Dimension r = 28, rank CP (X 0 ) e Dimension r = 35, rank CP (X 0 ) e Dimension r = 45, rank CP (X 0 ) e Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

42 Super-Symmetric Tensor Recovery Entries removed = 60% (sampling ratio = 40%): Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

43 Super-Symmetric Tensor Recovery Entries removed = 60% (sampling ratio = 40%): rank S CP (X 0 ) RelErr CPU rank M (X ) Dimension r = 8, rank S CP (X 0 ) e r = 12, rank S CP (X 0 ) e Dimension r = 8, rank S CP (X 0 ) e r = 20, rank S CP (X 0 ) e Dimension r = 15, rank S CP (X 0 ) e r = 25, rank S CP (X 0 ) e Dimension r = 15, rank S CP (X 0 ) e r = 30, rank S CP (X 0 ) e Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

44 Robust Tensor Recovery min rank CP (Y) + λ Z 0 s.t. Y + Z = F min M(Y) + λ Z 1 s.t. Y + Z = F Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

45 Robust Tensor Recovery min rank CP (Y) + λ Z 0 s.t. Y + Z = F min M(Y) + λ Z 1 s.t. Y + Z = F low-n-rank robust tensor recovery low-m-rank robust tensor recovery rank CP (Y 0 ) Rel Err All Rel Err LR Tucker rank Rel Err All Rel Err LR (rank M + (Y 0 ), rank M (Y 0 )) Dimension CP-rank e e-002 (3.85, 3.80, 3.90, 3.85) 8.48e e-007 (2,2) CP-rank e e-002 (5.30, 5.35, 5.15, 5.20) 7.76e e-007 (4,4) CP-rank e e-002 (10, 10, 10, 10) 8.63e e-006 (6,6) CP-rank e e-002 (10, 10, 10, 10) 7.78e e-006 (8,8) CP-rank e e-002 (10, 10, 10, 10) 8.61e e-006 (12,12) Dimension CP-rank e e-003 (5.15, 5.15, 5.25, 5.25) 8.50e e-007 (3,3) CP-rank e e-003 (15, 15, 15, 15) 8.57e e-007 (6,6) CP-rank e e-002 (15, 15, 15, 15) 8.13e e-007 (9,9) CP-rank e e-002 (15, 15, 15, 15) 7.54e e-007 (12,12) CP-rank e e-002 (15, 15, 15, 15) 8.11e e-007 (18,18) Dimension CP-rank e e-003 (20, 20, 20, 20) 8.64e e-007 (4,4) CP-rank e e-003 (20, 20, 20, 20) 7.37e e-007 (8,8) CP-rank e e-002 (20, 20, 20, 20) 8.59e e-007 (12,12) CP-rank e e-002 (20, 20, 20, 20) 8.26e e-007 (16,16) CP-rank e e-003 (20, 20, 20, 20) 8.59e e-007 (24,24) Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

46 Colored Video Completion: 50 frames The first row: frames of the original video; the second row: frames with 80% missing entries; the third row: recovered images using tensor completion. Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

47 Robust Video Recovery The first row are the 3 frames of the original video sequence. The second row are the recovered background, and the last row are the recovered foreground. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

48 Part III. Invariance of the Tucker Core Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

49 The Tucker Core Consider an N-way tensor X C I 1 I 2 I N. The equation X = G 1 A (1) 2 N A (N) G; A (1),..., A (N), is called a size-(j 1,..., J N ) Tucker decomposition of X, and G is called a core tensor associated with this decomposition, and A (n) is the n-th factorization matrix, where G C J 1 J 2 J N and matrices A (n) C I n J n, n = 1,..., N, and n is the mode-n (matrix) product. Moreover, a Tucker decomposition is said to be independent if each of the factorization matrix has full column rank. A Tucker decomposition is said to be orthonormal if each of the factorization matrix has orthonormal columns. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

50 The CP-Rank of the Tucker Core Theorem (Jiang, Yang, Z., 2015) For any given tensor X C I 1 I 2 I N with independent Tucker decomposition X = G; X (1),..., X (N), we have rank CP (X) = rank CP (G). Theorem (Jiang, Yang, Z., 2015) For any symmetric tensor X C I I I and its independent symmetric decomposition X = G s ; X, X,..., X, the core tensor G s C J J J is also symmetric, and rank S (G s ) = rank S (X). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

51 The Frobenius Norm Theorem (Jiang, Yang, Z., 2015) For any given tensor X C I 1 I 2 I N with independent Tucker decomposition X = G; X (1),..., X (N), we have that α G F X F β G F, where β = X (1) 2 X (2) 2 X (N) 2 and α = β/( N n=1 κ(x (n) )) with κ(x (n) ) being the condition number of the matrix X (n) for all n. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

52 The Tensor Schatten norm and p-quasi Norm For any tensor X C I 1 I 2 I N, the tensor p-quasi norm for 0 < p 1 of X is defined as X p min { r λ s p s=1 1/p : X = r λ s x (1) s x (2) s... x (N) s=1 s, } x (n) s = 1, s = 1, 2,..., r, n = 1, 2,..., N Theorem For any given tensor X C I 1 I 2 I N with independent Tucker decomposition X = G; X (1),..., X (N), we have α G p X p β G p, where β = X (1) 2 X (2) 2 X (N) 2 and α = β/( N n=1 κ(x (n) )) with κ(x (n) ) being the condition number of matrix X (n) for all n. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

53 The Border Rank For an N-way tensor X C I 1 I 2 I N, its border rank, denoted by rank B (X), is defined as rank B = min { r ɛ > 0, E C I 1 I 2 I N, s.t. E F ɛ, rank CP (X + E) r }. Theorem For any given tensor X C I 1 I 2 I N with independent Tucker decomposition X = G; X (1),..., X (N), we have rank B (X) = rank B (G). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

54 The Z-Eigenvalue For an N-way symmetric tensor T R I I, if there exists a number λ R and a nonzero vector x R I such that T (x (N 1) ) = λx, x T x = 1. Then λ is called the Z-eigenvalue of T, and x is called the corresponding Z-eigenvector. Theorem For any given N-way symmetric tensor T R I I I with exact independent symmetric Tucker decomposition T = G s ; X, X,..., X. Construct Ĝ s = G s 1 (X T X) 1/2 2 N (X T X) 1/2. Then any Z-eigenvalues of Ĝ s are also Z-eigenvalues of T while any non-zero Z-eigenvalue of T are also Z-eigenvalues of Ĝ s. Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

55 Numerical Experiments with the Z-Eigenvalue Computation Example 1 (Xie and Chang, 2014): X(x 4 ) = (x 1 + x 2 + x 3 + x 4 ) 4 + (x 2 + x 3 + x 4 + x 5 ) 4 R Examples 2 and 3 (Nie and Wang, 2014): X i jk = ( 1)i i with n = 5 and n = 8 respectively. + ( 1) j j + ( 1)k, k Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

56 Example 4 (Cui, Dai and Nie, 2014): X i1 i 2 i 3 i 4 = tan(i 1 ) + tan(i 2 ) + tan(i 3 ) + tan(i 4 ). Example 5 (Cui, Dai and Nie, 2014): X i1 i 2 i 3 i 4 i 5 = ln(i 1 ) + ln(i 2 ) + ln(i 3 ) + ln(i 4 ) + ln(i 5 ). Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

57 The Savings Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

58 The Savings C. Cui, Y. Dai, and J. Nie, All real eigenvalues of symmetric tensors, SIAM J. Math. Anal., 35: , Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36

59 The Savings C. Cui, Y. Dai, and J. Nie, All real eigenvalues of symmetric tensors, SIAM J. Math. Anal., 35: , Example Order Size Core Size CPU for Tensor CPU for Core s 3.29s s 2.55s s 2.66s s 4.25s s 5.23s Table: Computational Time Comparison Shuzhong Zhang (ISyE@UMN) Low-Rank Completion and Invariance of the Tucker Core January, / 36

60 Thank you! Shuzhong Zhang Low-Rank Completion and Invariance of the Tucker Core January, / 36