Robust Tensor Completion and its Applications. Michael K. Ng Department of Mathematics Hong Kong Baptist University

Transcription

1 Robust Tensor Completion and its Applications Michael K. Ng Department of Mathematics Hong Kong Baptist University

2 Example

3 Low-dimensional Structure Data in many real applications exhibit low-dimensional structures due to local regularities, global symmetries, repetitive patterns, redundant sampling,... (low-dimensional structure low-rank data matrices)

4 Example Customer/Item I II III IV A 5 1?? B? 2 3? C?? 4 2 D 1???..... For example (Netflix Challenge 2009), it is about 0.5 million users and about 18,000 movies Matrix Completion min X rank(x) subject to P Ω(X) = P Ω (M)

5 Example Matrix RPCA min X rank(x) + λ E 0 subject to X + E = M

6 Example Robust Matrix Completion min X rank(x) + λ E 0 subject to P Ω (X + E) = P Ω (M)

7 Low Rank Matrix Recovery Matrix Completion Matrix RPCA min X rank(x) subject to P Ω(X) = P Ω (M) min X rank(x) + λ E 0 subject to M = X + E Robust Matrix Completion min X rank(x) + λ E 0 subject to P Ω (M) = P Ω (X + E)

8 Singular Value Decomposition X = UΣV T U and V are orthogonal matrices and Σ is a diagonal matrix containing the singular values of X. The number of singular values of X is the rank of X.

9 Low Rank Matrix Recovery Matrix Completion Matrix RPCA min X X subject to P Ω (X) = P Ω (M) min X X + λ E 1 subject to M = X + E Robust Matrix Completion min X X + λ E 1 subject to P Ω (M) = P Ω (X + E) Nuclear norm : sum of singular values (convex envelop of rank)

10 Low Rank Matrix Recovery Results (RPCA) Candes, E. J., Li, X., Ma, Y., and Wright, J. Journal of the ACM, 58(3):173, (Matrix Completion) Recht, B. Journal of Machine Learning Research, 12(4): , (Matrix Completion) Chen, Y. IEEE Transactions on Information Theory, 61(5): , many papers...

11 Application: Background and Foreground Separation in Video (spatial dimensions + time)

12 Application: Color Images Completion and Denoising (spatial dimensions + RGB)

13 Application: Hyperspectral Images Completion and Denoising (spatial dimensions + frequencies)

14 Low Rank Tensor Recovery Data are usually in multi-dimensional array. Vectorization probably break the inherent structures and correlations in the original data.

15 Low Rank Tensor Recovery Tensor Completion min X rank(x ) subject to P Ω(X ) = P Ω (M) Tensor Robust PCA min X rank(x ) + λ E 0 subject to M = X + E Robust Tensor Completion min X rank(x ) + λ E 0 subject to P Ω (M) = P Ω (X + E)

16 Tensor Decomposition CANDECOMP/PARAFAC Decomposition: X = r λ i a i,1 a i,m i=1 The minimal value of r is called the rank of A.

17 Tensor Decomposition Tucker Decomposition: X = r 1 i 1 =1 X = G A 1 A 2 A m r m i m=1 g i1,i 2,,i m a i 1,1 a im,m It can be obtained by using singular value decomposition to each unfolded matrix X ij from X. The Tucker rank is (rank(x 1 ), rank(x 2 ),, rank(x m )) = (r 1, r 2,, r m ).

18 Low Rank Tensor Recovery Tensor Completion min X rank(x ) subject to P Ω(X ) = P Ω (M) Tensor Robust PCA min X rank(x ) + λ E 0 subject to M = X + E Robust Tensor Completion min X rank(x ) + λ E 0 subject to P Ω (M) = P Ω (X + E)

19 Low Rank Tensor Recovery CP decomposition/rank cannot be computed efficiently Matrix rank can be replaced by matrix nuclear norm (the sum of singular values), it is a convex envelope Replace Tucker rank by the sum of nuclear norms of unfolding tensors, interdependent matrix trace norm is involved The use of the sum of nuclear norms of unfolding matrices of a tensor may be challenged since it is suboptimal 1 The tensor trace norm (the average of trace norms of unfolding matrices) is not a tight convex relaxation of the tensor rank (the average rank of unfolding matrices) 2 1 C. Mu, B. Huang, J. Wright, and D. Goldfarb. Square deal: Lower bounds and improved relaxations for tensor recovery. In ICML, pages 7381, B. Romera-Paredes and M. Pontil. A new convex relaxation for tensor completion. In Adv. Neural Inf. Process. Syst., pages , 2013.

20 t-svd Decomposition A third-order tensor of size n 1 n 2 n 3 can be viewed as an n 1 n 2 matrix of tubes which lie in the third-dimension. [Kilmer, M. E. and Martin, C. D. Linear Algebra & Its Applications, 435(3):641658, 2011]

21 t-svd Decomposition Definition: The t-product A B of A R n 1 n 2 n 3 and B R n 2 n 4 n 3 is a tensor C R n 1 n 4 n 3 whose (i, j)th tube is given by n 2 C(i, j, :) = A(i, k, :) B(k, j, :), k=1 where denotes the circular convolution between two tubes of same size. The tube at (i, k) position in A convolutes with the tube at (k, j) position in B. Both have sizes n 3. Put all the correlations at (i, j) position in C. The multiplication of between the scalars is replaced by circular convolution between the tubes.

22 t-svd Decomposition Definition: The identity tensor I R n n n 3 is defined to be a tensor whose first frontal slice I (1) is the n n identity matrix and whose other frontal slices I (i), i = 2,..., n 3 are zero matrices. Definition: The conjugate transpose of a tensor A R n 1 n 2 n 3 is the tensor A H R n 2 n 1 n 3 obtained by conjugate transposing each of the frontal slice and then reversing the order of transposed frontal slices 2 through n 3, i.e., (A H) (1) (A H) (i) = = ( A (1)) H, ( ) A (n H 3+2 i), i = 2,..., n3.

23 t-svd Decomposition Definition: A tensor Q R n n n 3 is orthogonal if it satisfies Q H Q = Q Q H = I, where I is the identity tensor of size n n n 3. Definition: A tensor A is called f-diagonal if each frontal slice A (i) is a diagonal matrix.

24 t-svd Decomposition For A R n 1 n 2 n 3, the t-svd of A is given by A = U S V H, where U R n 1 n 1 n 3 and V R n 2 n 2 n 3 are orthogonal tensors, and S R n 1 n 2 n 3 is a f-diagonal tensor, respectively. The entries in S are called the singular tubes of A.

25 t-svd Decomposition The tensor tubal-rank, denoted as rank t (A), is defined as the number of nonzero singular tubes of S, where S comes from the t-svd of A, i.e., rank t (A) = #{i : S(i, i, :) 0}. It can be shown that it is equal to max i rank(â (i) ) where  (i) is the i-th slice of  and  represents a third-order tensor obtained by taking the Discrete Fourier Transform (DFT) of all the tubes along the third dimension of A. Example of t-svd Decomposition

26 Hyperspectral Images

27 Tubal Rank 1 Approximation

28 Tubal Rank 5 Approximation

29 Tubal Rank 10 Approximation

30 Tubal Rank 20 Approximation

31 Tubal Rank 40 Approximation

32 Low Tubal Rank Tensor Recovery Tensor Completion min X rank(x ) subject to P Ω(X ) = P Ω (M) Tensor Robust PCA min X rank(x ) + λ E 0 subject to M = X + E Robust Tensor Completion min X rank(x ) + λ E 0 subject to P Ω (M) = P Ω (X + E)

33 TNN Definition: The tubal nuclear norm of a tensor A R n 1 n 2 n 3, denoted as A TNN, is the nuclear norm of all the frontal slices of Â. Theorem For any tensor X C n 1 n 2 n 3, X TNN is the convex envelope of the function n 3 i=1 rank(â(i) ) on the set {X X 1}.

34 Low Tubal Rank Tensor Recovery (Relaxation) Tensor Completion min X X TNN subject to P Ω (X ) = P Ω (M) Tensor Robust PCA min X X TNN + λ E 1 subject to M = X + E Robust Tensor Completion min X X TNN + λ E 1 subject to P Ω (M) = P Ω (X + E) Can we recover low-tubal-rank tensor from partial and grossly corrupted observations exactly?

35 Tensor Incoherence Conditions Assume that rank t (L 0 ) = r and its t-svd L 0 = U S V H. L 0 is said to satisfy the tensor incoherence conditions with parameter µ > 0 if µr max U H e i F, i=1,,n 1 n 1 max V H e j F j=1,,n 2 and (joint incoherence condition) U V H µr n 2, µr n 1 n 2 n 3.

36 Tensor Incoherence Conditions The column basis, denoted as e i, is a tensor of size n 1 1 n 3 with its (i, 1, 1)th entry equaling to 1 and the rest equaling to 0. The tube basis, denoted as ek, is a tensor of size 1 1 n 3 with its (1, 1, k)th entry equaling to 1 and the rest equaling to 0.

37 Low Rank Tensor Recovery Theorem Suppose L 0 R n 1 n 2 n 3 obeys tensor incoherence conditions, and the observation set Ω is uniformly distributed among all sets of cardinality m = ρn 1 n 2 n 3. Also suppose that each observed entry is independently corrupted with probability γ. Then, there exist universal constants c 1, c 2 > 0 such that with probability at least 1 c 1 (n (1) n 3 ) c 2, the recovery of L 0 with λ = 1/ ρn (1) n 3 is exact, provided that r c r n (2) µ(log(n (1) n 3 )) 2 and γ c γ where c r and c γ are two positive constants. n (1) = max{n 1, n 2 } and n (2) = min{n 1, n 2 }

38 Low Rank Tensor Recovery Theorem (Tensor Completion): Suppose L 0 R n 1 n 2 n 3 obeys tensor incoherence conditions, and m entries of L 0 are observed with locations sampled uniformly at random, then there exist universal constants c 0, c 1, c 2 > 0 such that if m c 0 µrn (1) n 3 (log(n (1) n 3 )) 2, L 0 is the unique minimizer to the convex optimization problem with probability at east 1 c 1 (n (1) n 3 ) c 2.

39 Low Rank Tensor Recovery Theorem (Tensor Robust PCA): Suppose L 0 R n 1 n 2 n 3 obeys tensor incoherence conditions and joint incoherence condition and E 0 has support uniformly distributed with probability γ. Then, there exist universal constants c 1, c 2 > 0 such that with probability at least 1 c 1 (n (1) n 3 ) c 2, (L 0, E 0 ) is the unique minimizer to the convex optimization problem with λ = 1/ n (1) n 3, provided that r c r n (2) µ(log(n (1) n 3 )) 2 and γ c γ where c r and c γ are two positive constants.

40 Convex Optimization Problem Input: X, Ω and λ. Initialize: L 0 = E 0 = Y 0 = 0, ρ = 1.1, µ 0 = 1e-4, µ max = 1e8. WHILE not converged 1. Update L k+1 by 2. Update P Ω (E k+1 ) by min L TNN + µk L + E k X + Yk 2 L 2 µ k ; F ( ) min λ P Ω (E) 1 + µk P Ω E + L k+1 X + Yk 2 E 2 µ k ; F 3. Update P Ω c (E k+1 ) by P Ω c (E k+1 ) = P Ω c (X L k+1 Y k /µ k ); 4. Update the multipliers Y k+1 by Y k+1 = Y k + µ k (L k+1 + E k+1 X ); 5. Update µ k+1 by µ k+1 = min(ρµ k, µ max ); 6. Check the convergence condition ENDWHILE Output: L

41 Phase Transition ρ (data observation) and γ (data corruption)

42 Application: Completion and Denoising ρ = 70% (data observation) and γ = 30% (data corruption)

43 Application: Completion and Denoising ρ = 70% (data observation) and γ = 30% (data corruption)

44 Application: Completion and Denoising For RPCA and RMC, we apply them on each channel with λ = 1/ n 1 ; For SNN, unfolding with three parameters suggested in the literature; For TRPCA, λ = 1/ n 1 n 3 ; For BM3D, standard denoising method using nonlocal information; For BM3D+, two-step method with BM3D and image completion using HaLRTC (tensor unfolding to matrix); For BM3D++, two-step method with BM3D and image completion using TNMM.

45 Video Background Modeling Background: Low-Tubal-Rank Component and Moving Objects: Sparse Component

46 Video Background Modeling

47 Traffic Data Estimation Traffic flow data such as traffic volumes, occupancy rats and flow speeds are usually contaminated by missing values and outliers due to the hardware or software malfunctions. Performance Measurement System (PeMS) pems.dot.ca.gov Third-order tensor (day) x (time) x (week) of traffic volume

48 Traffic Data Estimation

49 Transform Orientation: Example 3rd direction 1st direction 2nd direction

50 The Correction Model

51 The Corrected Model Issue: The nuclear norm minimization of a matrix may be challenged under general sampling distribution. Salakhutdinov et al. 3 showed that when certain rows and/or columns were sampled with high probability, the matrix nuclear norm minimization may fail in the sense that the number of observations required for recovery was much more than the setting of most matrix completion problems. Miao et al. proposed a rank-corrected model for low-rank matrix recovery with fixed basis coefficients 4. 3 R. Salakhutdinov and N. Srebro. Collaborative filtering in a non-uniform world: Learning with the weighted trace norm. In Adv. Neural Inform. Process. Syst., pages , W. Miao, S. Pan, and D. Sun. A rank-corrected procedure for matrix completion with fixed basis coefficients. Math. Program., 159(1):289338, 2016.

52 The Corrected Method For any given index set Ω {1, 2,..., n 1 } {1, 2,..., n 2 } {1, 2,..., n 3 }, we define the sampling operator D Ω : R n 1 n 2 n 3 R Ω by D Ω (X ) = ( E ijk, X ) T (i,j,k) Ω, where Ω denotes the number of entries in Ω. Let X 0 R n 1 n 2 n 3 be an unknown true tensor. The observed model can be described in the following form: y = D Ω (X 0 ) + σε, where y = (y 1, y 2,..., y m ) T R m and ε = (ε 1, ε 2,..., ε m ) T R m are the observation vector and the noise vector, respectively, ε i are the independent and identically distributed (i.i.d.) noises with E(ε i ) = 0 and E(ε 2 i ) = 1, and σ > 0 controls the magnitude of noise.

53 The Corrected Method Assumption: Each entry is sampled with positive probability, i.e., there exists a positive constant κ 1 1 such that It implies p ijk n 1 n 2 n 3 E( E, X 2 ) = i=1 j=1 k=1 1 κ 1 n 1 n 2 n 3. p ijk x 2 ijk 1 κ 1 n 1 n 2 n 3 X 2 F.

54 The Corrected Method In the matrix case, the nuclear norm penalization may fail when some columns or rows are sampled with very high probability. In the third-order tensor, we also need to avoid this case that each fiber is sampled with very high probability. Let n 1 n 2 n 3 R jk = p ijk, C ik = p ijk, T ij = p ijk, i=1 j=1 k=1 Assumption: There exists a positive constant κ 2 1 such that max {R jk, C ik, T ij } i,j,k κ 2 min{n 1, n 2, n 3 }.

55 The Corrected Method min 1 ( ) 2m y D Ω(X ) 2 + µ X TNN F (X m ), X s.t. X c, where the spectral function F : R n 1 n 2 n 3 R n 1 n 2 n 3 is given as follows: F (X m ) := U Σ V H, associated with Σ = ifft( M, [ ], 3) with M ( ( )) (i) = f (Ŝ (i) ) := Diag f diag(ŝ (i) ), f is defined by f i (x) := { ( ) φ xi x, if x 0, 0, otherwise, and the scalar function φ : R R, is defined by φ(z) = (1 + ε τ z τ ) z τ + ε τ.

56 The Corrected Method The correction function F is used to get a lower tubal rank solution. For the small singular values of the frontal slices in the Fourier domain, we would like to penalize more in the correction procedure. Then these small singular values will approximate to zero in the next correction procedure. In this case, the model can generate a lower tubal rank solution by the correction method.

57 The Corrected Method Theorem Suppose the two assumptions hold. Let τ > 1 be given. Then, for m ñn 3 log 3 (n 1 n 3 + n 2 n 3 )/κ 2, there exists constants C, C 1 > 0 such that X c X 0 2 F n 1n 2 κ 2 1 κ 2 log((n 1 + n 2 )n 3 ) n 1 n 2 n 3 mñ ( ( 32C1 2 2r ) 2τ + α 2 m σ 2 + τ 4096 Cc 2( τ( ) 2r + α m ) ) 2 τ 1 with probability at least 1 2 n 1 +n 2 +n 3, where α m = Ũ1 ṼT 1 F (X m) F. Here Ũ1, ṼT 1 are the associated orthogonal tensors in t-svd of X 0.

58 The Symmetric Gauss-Seidel Multi-Block ADMM Let U(X ) := {X X c}. By introducing z = y D Ω (X ) and X = S, the model is given by ( ) 1 min 2m z 2 + µ X TNN F (X m ), X + δ U (S) s.t. z = y D Ω (X ), X = S. Since the TNN is the dual norm of the tensor spectral norm, its Lagrangian dual is given as follows: max u,w m 2 u 2 + u, y δ U ( W) s.t. µf (X m ) + D Ω (u) + W µ.

59 The Symmetric Gauss-Seidel Multi-Block ADMM Let Z := µf (X m ) D Ω (u) + W and X(X ) := {X X µ}. min u,w,z m 2 u 2 u, y + δ U ( W) + δ X(Z) s.t. Z = µf (X m ) + D Ω (u) + W. The augmented Lagrangian function is defined by L(u, W, Z, X ) := m 2 u 2 u, y + δ U ( W) + δ X(Z) X, Z µf (X m ) D Ω (u) W + β 2 Z µf (X m) D Ω (u) W 2 F, where β > 0 is the penalty parameter and X is the Lagrangian multiplier.

60 The Symmetric Gauss-Seidel Multi-Block ADMM The iteration system of sgs-admm is described as follows: { } u k+ 1 2 = arg min L(u, W k, Z k, X k ), u { } W k+1 = arg min L(u k+ 1 2, W, Z k, X k ), W { } u k+1 = arg min L(u, W k+1, Z k, X k ), u { } Z k+1 = arg min L(u k+1, W k+1, Z, X k ), Z ( X k+1 = X k γβ Z k+1 µf (X m ) D Ω (uk+1 ) W k+1), where γ (0, (1 + 5)/2) is the step-length.

61 The Symmetric Gauss-Seidel Multi-Block ADMM The optimal solution with respect to u is given explicitly by u = 1 ( )) (y D Ω X + β(µf (X m ) + W Z). m + β The optimal solution with respect to W is given explicitly by ( ) W k+1 = Prox 1 1 β δ U β X k + µf (X m ) + D Ω (uk+ 1 2 ) Z k ( ) = 1 β X k + µf (X m ) + D Ω (uk+ 1 2 ) Z k ( )) + 1 β Prox βδ U (β 1 β X k + µf (X m ) + D Ω (uk+ 1 2 ) Z k.

62 The Symmetric Gauss-Seidel Multi-Block ADMM For the subproblem with respect to Z, it is a projection onto X, which has a closed-form solution. Theorem For any Y R n 1 n 2 n 3 and ρ > 0, let Y = U S V H be the t-svd. Then the optimal solution X of the following problem is given by min X R n 1 n 2 n 3 { X Y 2 F, X ρ} X = U S ρ V H, where S ρ = ifft(min{s, ρ}, [], 3).

63 The Symmetric Gauss-Seidel Multi-Block ADMM The optimal solution with respect to Z in (1) is given by ) Z k+1 = Prox 1 (µf β δ (X X m ) + D Ω (uk+1 ) + W k β X k+1 = U k+1 S k+1 µ (V k+1 ) T, where S k+1 µ = ifft(min{s k+1, µ}, [ ], 3) and µf (X m ) D Ω (uk+1 ) + W k β X k+1 = U k+1 S k+1 (V k+1 ) T.

64 The Symmetric Gauss-Seidel Multi-Block ADMM Theorem The optimal solution set is nonempty and compact. Only two blocks with respect to W, Z are nonsmooth and other blocks are quadratic. Theorem Suppose that β > 0 and γ (0, (1 + 5)/2). Let the sequence {(W k, u k, Z k, X k )} be generated by the algorithm. Then {(W k, u k, Z k )} converges to an optimal solution and {X k } converges to an optimal solution of the dual problem.

65 Numerical Examples Table: Relative errors of the TNN and CTNN with different tensors, tubal ranks, and sampling ratios for low-rank tensor recovery. Tensor r σ SR TNN CTNN-1 CTNN-2 CTNN e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-3

66 Color Image

67 Color Image

68 Other t-svds

69 Revisit t-svd We use X C m 1 m 2 m 3 to represent the discrete Fourier transform of X C m 1 m 2 m 3 along each tube, i.e., X = fft(x, [ ], 3). The block circulant matrix is defined as bcirc(x ) := X (1) X (m 3) X (2) X (2) X (1) X (3) X (m 3) X (m 3 1) X (1) The block diagonal matrix and the corresponding inverse operator are defined as X (1) X (2) bdiag(x ) :=..., X (m3)

70 Revisit t-svd Theorem bdiag( X ) = (F m3 I m1 )bcirc(x )(F H m 3 I m2 ), where denotes the Kronecker product, F m3 matrix and I m is an m m identity matrix. is an m 3 m 3 DFT

71 Revisit t-svd The unfold and fold operators in t-svd are defined as X (1) X (2) unfold(x ) :=, fold(unfold(x )) = X.. X (m 3) Given X C m 1 m 2 m 3 and Y C m 2 m 4 m 3, the t-product X Y is a third-order tensor of size m 1 m 4 m 3 Z = X Y := fold(bcirc(x )unfold(y)). Since the corresponding block circulant matrices can be diagonalized by DFT, the DFT based t-svd can be efficiently implemented via fast Fourier transform (fft).

72 Cosine-Transform based t-svd The first work is given by E. Kernfeld, M. Kilmer and S. Aeron, Tensor tensor products with invertible linear transforms, LAA, Vol 485, pp (2015). We define the shift of tensor A = fold σ(a) = fold A (2) A (3). A (m 3) O. A (1) A (2). A (m 3) as Any tensor X can be uniquely divided into A + σ(a).

73 Cosine-Transform based t-svd We use X R m 1 m 2 m 3 to represent the DCT along each tube of X, i.e., X = dct(x, [ ], 3) = dct(a + σ(a), [ ], 3). We define the block Toeplitz matrix of A as A (1) A (2) A (m 3 1) A (m 3) A (2) A (1) A (m 3 2) A (m 3 1) bt(a) := A (m 3 1) A (m 3 2) A (1) A (2) A (m 3) A (m 3 1) A (2) A (1) The block Hankel matrix is defined as A (2) A (3) A (m 3) O A (3) A (4) O A (m 3) bh(a) := A (m 3) O A (4) A (3) O A (m 3) A (3) A (2).

74 Cosine-Transform based t-svd The block Toeplitz-plus-Hankel matrix of A is defined as btph(a) := bt(a) + bh(a). The block Toeplitz-plus-Hankel matrix can be diagonalized. Theorem bdiag( X ) = (C m3 I m1 )btph(a)(c T m 3 I m2 ), where denotes the Kronecker product, C m3 matrix. is an m 3 m 3 DCT

75 Cosine-Transform based t-svd Definition: Given X C m 1 m 2 m 3 and Y C m 2 m 4 m 3, the t-product X Y is a third-order tensor of size m 1 m 4 m 3 Z = X Y := fold(btph(a)unfold(y)), where X = A + σ(a).

76 Cosine-Transform based t-svd Theorem Given a tensor X R m 1 m 2 m 3, the DCT-based t-svd of X is given by X = U dct S dct V H, where U R m 1 m 1 m 3,V R m 2 m 2 m 3 are orthogonal tensors, S R m 1 m 2 m 3 is a f-diagonal tensor, and V H is the tensor transpose of V.

77 Cosine-Transform based t-svd Table: The time cost of t-svd and DCT-based t-svd on the random tensors of different size. size 100*100* *100* *200* *400*100 FFT SVD after FFT original t-svd DCT SVD after DCT new t-svd

78 Video Examples

79 Table: PSNR, SSIM, and time of two methods in video completion. In brackets, they are the time required for transformation and time required for performing SVD. The best results are highlighted in bold. video akiyo suzie salesman SR metric TNN-F TNN-C TNN-F TNN-C TNN-F TNN-C PSNR SSIM time PSNR SSIM time PSNR SSIM time

80 Video Examples

81 Transform-based t-svd Fourier-Transform based t-svd Z = X fft Y = fold(bcirc(x )unfold(y)) The DFT based t-svd can be efficiently implemented via fast Fourier transform (fft). Cosine-Transform based t-svd Z = X dct Y = fold(btph(a)unfold(y)) The DCT based t-svd can be efficiently implemented via fast cosine transform (dct).

82 Transform-based t-svd Fourier-Transform based t-svd ( ) Z = X fft Y = fold[unfold( ˆX fft ). unfold(ŷ fft )] The DFT based t-svd can be efficiently implemented via fast Fourier transform (fft). Cosine-Transform based t-svd ( ) Z = X dct Y = fold[unfold( ˆX dct ). unfold(ŷ dct )] The DCT based t-svd can be efficiently implemented via fast cosine transform (dct). ifft idct

83 Transform-based t-svd The first work is given by E. Kernfeld, M. Kilmer and S. Aeron, Tensor tensor products with invertible linear transforms, LAA, Vol 485, pp (2015). We generalize tensor singular value decomposition by using other unitary transform matrices instead of discrete Fourier/cosine transform matrix. The motivation is that a lower transformed tubal tensor rank may be obtained by using other unitary transform matrices than that by using discrete Fourier/cosine transform matrix, and therefore this would be more effective for robust tensor completion.

84 Transform-based t-svd Let Φ be the unitary transform matrix with ΦΦ H = Φ H Φ = I. Â Φ represents a third-order tensor obtained via multiplying by Φ on all tubes along the third dimension of A. The Φ-product of A C n 1 n 2 n 3 and B C n 2 n 4 n 3 is a tensor C C n 1 n 4 n 3, which is given by ( ) C = A Φ B = fold[unfold( ˆX Φ ). unfold(ŷ Φ )] Φ H.

85 Transform-based t-svd Theorem Suppose that A C n 1 n 2 n 3. Then A can be factorized as follows: A = U Φ S Φ V H, where U C n 1 n 1 n 3, V C n 2 n 2 n 3 are unitary tensors with respect to Φ-product, and S C n 1 n 2 n 3 is a diagonal tensor.

86 Transform-based t-svd Definition: The transformed tubal multi-rank of a tensor A C n 1 n 2 n 3 is a vector r R n 3 with its i-th entry as the rank of the i-th frontal slice of  Φ, i.e., r i = rank(â (i) Φ ). The transformed tubal tensor rank, denoted as rank tt (A), is defined as the number of nonzero singular tubes of S, where S comes from the tt-svd of A = U Φ S Φ V H. Definition: The transformed tubal nuclear norm of a tensor A C n 1 n 2 n 3, denoted as A TTNN, is the sum of the nuclear norms of all the frontal slices of  Φ, i.e., n 3 A TTNN = Â(i) Φ. i=1

87 Transform-based t-svd Theorem For any tensor X C n 1 n 2 n 3, X TTNN is the convex envelope of the function n 3 i=1 rank(â(i) ) on the set {X X 1}. Φ

88 Transform-based t-svd min L TTNN + λ E 1, s.t., P Ω (L + E) = P Ω (X ), L, E where λ is a penalty parameter and P Ω is a linear projection such that the entries in the set Ω are given while the remaining entries are missing.

89 Transform-based t-svd Assume that rank tt (L 0 ) = r and its skinny tt-svd is L 0 = U Φ S Φ V H. L 0 is said to satisfy the transformed tensor incoherence conditions with parameter µ > 0 if and max U H Φ ei F i=1,...,n 1 max j=1,...,n 2 V H Φ e j F U Φ V H µr, n 1 µr, n 2 µr n 1 n 2 n 3, where ei and ej are the tensor basis with respect to Φ.

90 Transform-based t-svd Theorem Suppose that L 0 C n 1 n 2 n 3 obeys transformed tensor incoherence conditions, and the observation set Ω is uniformly distributed among all sets of cardinality m = ρn 1 n 2 n 3. Also suppose that each observed entry is independently corrupted with probability γ. Then, there exist universal constants c 1, c 2 > 0 such that with probability at least 1 c 1 (n (1) n 3 ) c 2, the recovery of L 0 with λ = 1/ ρn (1) n 3 is exact, provided that r c r n (2) µ(log(n (1) n 3 )) 2 and γ c γ, where c r and c γ are two positive constants.

91 Numerical Illustration Table: The transformed tubal ranks of randomly generated ten tensors. Transform/Tensor #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 level-1 Haar level-2 Haar Fourier

92 Numerical Illustration Table: The relative errors of tensor completion for Tensors #1 and #2 with sampling ratios. ρ Tensor #1 Tensor #2 level-1 Haar level-2 Haar Fourier level-1 Haar level-2 Haar Fourier e e e e e e e e e e e e e e e e e e-1

93 Numerical Illustration Table: The relative errors of robust tensor completion for Tensors #1 and #2 with sampling ratios and noise levels. ρ γ Tensor #1 Tensor #2 level-1 Haar level-2 Haar Fourier level-1 Haar level-2 Haar Fourier e e e e e e e e e0 2.26e e e e e e0 3.67e e e e e e0 2.63e e e e e e0 3.35e e e e e e0 1.77e e e0

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97 Data-Dependent Transform Theorem Let A be a given tensor with size n 1 n 2 n 3 and A be its unfolding matrix along the third-dimension. Suppose rank(a) = k. Then a global optimal solution of the following problem min rank(b)=k,φ ΦA B 2 F is given by Φ = U H and B = ΣV H. s.t. Φ H Φ = ΦΦ H = I,

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101 Concluding Remarks Other tensor data sets Theory and Algorithms to be studied