Successive Partial-Symmetric Rank-One Algorithms for Almost Unitarily Decomposable Conjugate Partial-Symmetric Tensors

Fu, Taoran; Fan, Junxuan

doi:10.1007/s40305-018-0194-6

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…By Theorem 3.2, both H(T ) and iS(T ) can be decomposed into a sum of rank-one CPS tensors as in (7), with coefficients being real numbers. Therefore, T = H(T ) + (−i)iS(T ) can be decomposed into a sum of rank-one CPS tensors as in (9), with coefficients being complex numbers. Corollary 3.3 also provides an alternative definition of a PS tensor via a complex linear combination of rank-one CPS tensors, i.e.,…”

Section: Corollary 33 An Even-order Tensor T ∈ C N 2d Is Ps If and On...mentioning

confidence: 99%

“…In fact, (10) can be immediately obtained from ( 9) by absorbing each λ j into a j ⊗d . This makes the decomposition (9) interesting as it links the first half and the last half modes of a PS tensor, which is not obvious either from Definition 2.1 or the decomposition (10). Even for d = 1, (9) reduces to that any complex matrix A ∈ C n 2 can be written as A = m j=1 λ j a j a j H with λ j ∈ C and a j ∈ C n for j = 1, .…”

Section: Corollary 33 An Even-order Tensor T ∈ C N 2d Is Ps If and On...mentioning

confidence: 99%

See 1 more Smart Citation

On decompositions and approximations of conjugate partial-symmetric complex tensors

Fu¹,

Jiang²,

Li³

2018

Preprint

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Conjugate partial-symmetric (CPS) tensors are the high-order generalization of Hermitian matrices. As the role played by Hermitian matrices in matrix theory and quadratic optimization, CPS tensors have shown growing interest recently in tensor theory and optimization, particularly in many application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, rank-one decompositions and approximations, as well as their applications. The analysis is conducted along side with a more general class of complex tensors called partial-symmetric tensors. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an alternative definition of CPS tensors via linear combinations of rank-one CPS tensors. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon's conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximation of CPS tensors. Numerical experiments from various data are performed to justify the capability of our methods.

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Section: Corollary 33 An Even-order Tensor T ∈ C N 2d Is Ps If and On...mentioning

confidence: 99%

Section: Corollary 33 An Even-order Tensor T ∈ C N 2d Is Ps If and On...mentioning

confidence: 99%

On decompositions and approximations of conjugate partial-symmetric complex tensors

Fu¹,

Jiang²,

Li³

2018

Preprint

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“…For instance, every symmetric complex form generated by a CPS tensor is real-valued and all the eigenvalues of a CPS tensor are real [24]. In contrast to the many efforts on the optimization aspect [12,13,19,23,40,45] of CPS tensors, the current paper aims for their decompositions, ranks and approximations, which are important topics for high-order tensors. As we all know that the generalization of matrices to high-order tensors has led to interesting new findings as well as keeping many nice properties, CPS tensors, as a generalization of Hermitian matrices in terms of the high order and a generalization of real symmetric tensors in terms of the complex field, should also be expected to behave in that sense.…”

Section: Introductionmentioning

confidence: 99%

“…12 If T ∈ ℂ n 2d cps and a permutation ∈ Π(1, … , 2d) satisfies then M (T) is a CPS (Hermitian) matrix.…”

mentioning

confidence: 99%

On decompositions and approximations of conjugate partial-symmetric tensors

Fu¹,

Jiang

2021

Calcolo

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Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon’s conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods.

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“…Zhang et al [9] first proved the successive algorithm exactly recovers the symmetric and orthogonal decomposition of the underlying real symmetrically and orthogonally decomposable tensors. Fu et al [10] showed that SROA algorithm can exactly recover unitarily decomposable CPS tensors. We offer the theoretical guarantee of SROA algorithm for our matrix decomposition model of CPS tensor tensors.…”

Section: Introductionmentioning

confidence: 99%

The low-rank approximation of fourth-order partial-symmetric and conjugate partial-symmetric tensor

Sabir¹,

Huang²,

Yang³

2021

Preprint

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We present an orthogonal matrix outer product decomposition for the fourth-order conjugate partial-symmetric (CPS) tensor and show that the greedy successive rank-one approximation (SROA) algorithm can recover this decomposition exactly. Based on this matrix decomposition, the CP rank of CPS tensor can be bounded by the matrix rank, which can be applied to low rank tensor completion. Additionally, we give the rankone equivalence property for the CPS tensor based on the SVD of matrix, which can be applied on the rank-one approximation for CPS tensors.

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Successive Partial-Symmetric Rank-One Algorithms for Almost Unitarily Decomposable Conjugate Partial-Symmetric Tensors

Cited by 5 publications

References 12 publications

On decompositions and approximations of conjugate partial-symmetric complex tensors

On decompositions and approximations of conjugate partial-symmetric complex tensors

On decompositions and approximations of conjugate partial-symmetric tensors

The low-rank approximation of fourth-order partial-symmetric and conjugate partial-symmetric tensor

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