### Citations

1799 |
Independent component analysis – a new concept
- Comon
- 1994
(Show Context)
Citation Context ...de applications in wireless communication systems, magnetic resonance imaging, signal and image processing, data analysis, higher order statistics, as well as independent component analysis [2], [3[, =-=[4]-=-, [6[, [7[, [10[, [12], [14], [15], [17[, [19[, [21], [23], [26[. A basic question for the best rank-one approximation problem is whether there exists a positive lower bound for the quotient of the be... |

638 |
W.C.: Iterative solution of nonlinear equations in several variables
- Ortega, Rheinboldt
- 1970
(Show Context)
Citation Context ...oximation of any tensor in V and the norm of that tensor. In the finite dimensional case. Si is closed. Then, by (2.2), we see that a(-) is also a norm of V. By (2.4) and the norm equivalence theorem =-=[18]-=-, we have (2.7) App(V) > 0. Thus, in the finite dimensional case, (2.6) provides an upper bound for the quotient of the residual of the best rank-one approximation of any tensor «4 in V and the norm o... |

138 |
Some remarks on greedy algorithms
- DeVore, Temlyakov
- 1996
(Show Context)
Citation Context ...ximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm [1], [9[, =-=[8]-=-, [24]. In the next section, we show that such a positive lower bound exists. We call it the best rank-one approximation ratio of that tensor space. In section 3, we give a positive lower bound for th... |

76 | On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors
- Kofidis, Regalia
- 2002
(Show Context)
Citation Context ...reless communication systems, magnetic resonance imaging, signal and image processing, data analysis, higher order statistics, as well as independent component analysis [2], [3[, [4], [6[, [7[, [10[, =-=[12]-=-, [14], [15], [17[, [19[, [21], [23], [26[. A basic question for the best rank-one approximation problem is whether there exists a positive lower bound for the quotient of the best rank-one approximat... |

40 |
Estimating crossing fibers: A tensor decomposition approach
- Schultz, Seidel
(Show Context)
Citation Context ...ic resonance imaging, signal and image processing, data analysis, higher order statistics, as well as independent component analysis [2], [3[, [4], [6[, [7[, [10[, [12], [14], [15], [17[, [19[, [21], =-=[23]-=-, [26[. A basic question for the best rank-one approximation problem is whether there exists a positive lower bound for the quotient of the best rank-one approximation of a tensor and the norm of that... |

17 | Rank-1 approximation of higher-order tensors - Zhang, Golub |

15 |
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach
- Falcó, Nouy
(Show Context)
Citation Context ...nite dimensional biquadratic tensor space. Some numerical results are given in section 6. Four open questions are raised in section 7. 2. General discussion. The following discussion is borrowed from =-=[9]-=- and was suggested by a referee. Let Vj be separable Hilbert spaces with inner product (•,-)j ÎOT j = 1, ... ,m. Consider the tensor product Hilbert space V = 0 ! ^ j F^ (or the subspace of symmetric ... |

13 |
Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems
- So
- 2011
(Show Context)
Citation Context ...ric and general tensors, there are also various partially symmetric tensors. Among partially symmetric tensors, biquadratic tensors have received much attention in recent years [5], [11], [16], [20], =-=[22]-=-, [25], [27]. An (n X p)-dimensional biquadratic tensor A has the form A = (aijki), where i,j = l,...,n; k,l=l,...,p; 2 < n < p, with symmetric property ajj^i = aju^i = üijik for any i, j , k, and /. ... |

11 | Higherorder power method–Application in independent component analysis - Lathauwer, Comon, et al. - 1995 |

11 | Regalia, Tensor displacement structures and polyspectral matching, in Fast Reliable Algorithms for Matrices with Structure - Grigorascu, A - 1999 |

10 |
A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor
- Wang, Qi, et al.
- 2009
(Show Context)
Citation Context ...d general tensors, there are also various partially symmetric tensors. Among partially symmetric tensors, biquadratic tensors have received much attention in recent years [5], [11], [16], [20], [22], =-=[25]-=-, [27]. An (n X p)-dimensional biquadratic tensor A has the form A = (aijki), where i,j = l,...,n; k,l=l,...,p; 2 < n < p, with symmetric property ajj^i = aju^i = üijik for any i, j , k, and /. We use... |

9 |
A tensor product matrix approximation problem in quantum physics
- Dahl, Leinass, et al.
- 2007
(Show Context)
Citation Context ...or space. Beside symmetric and general tensors, there are also various partially symmetric tensors. Among partially symmetric tensors, biquadratic tensors have received much attention in recent years =-=[5]-=-, [11], [16], [20], [22], [25], [27]. An (n X p)-dimensional biquadratic tensor A has the form A = (aijki), where i,j = l,...,n; k,l=l,...,p; 2 < n < p, with symmetric property ajj^i = aju^i = üijik f... |

8 |
On the best rank-1 approximation to higher-order symmetric tensors
- Ni, WANG
- 1993
(Show Context)
Citation Context ...nication systems, magnetic resonance imaging, signal and image processing, data analysis, higher order statistics, as well as independent component analysis [2], [3[, [4], [6[, [7[, [10[, [12], [14], =-=[15]-=-, [17[, [19[, [21], [23], [26[. A basic question for the best rank-one approximation problem is whether there exists a positive lower bound for the quotient of the best rank-one approximation of a ten... |

5 | Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints
- Zhang, Ling, et al.
(Show Context)
Citation Context ...ral tensors, there are also various partially symmetric tensors. Among partially symmetric tensors, biquadratic tensors have received much attention in recent years [5], [11], [16], [20], [22], [25], =-=[27]-=-. An (n X p)-dimensional biquadratic tensor A has the form A = (aijki), where i,j = l,...,n; k,l=l,...,p; 2 < n < p, with symmetric property ajj^i = aju^i = üijik for any i, j , k, and /. We use B„p t... |

1 |
On the convergence ofa greedy rank-one update algorithm for a class of linear systems
- CHINESTA, FALCÓ
- 1999
(Show Context)
Citation Context ...-one approximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm =-=[1]-=-, [9[, [8], [24]. In the next section, we show that such a positive lower bound exists. We call it the best rank-one approximation ratio of that tensor space. In section 3, we give a positive lower bo... |

1 | A viodified Newtons method for best rank-one approximation to tensors - CiiANG, Y |

1 | On the best rank-1 and rank-(R¡ ,rt¿ R^i) approximation of higher-order tensor - MOOR |

1 |
Conditions for strong ellipticity and M-eigenvalues
- unknown authors
(Show Context)
Citation Context ...symmetric and general tensors, there are also various partially symmetric tensors. Among partially symmetric tensors, biquadratic tensors have received much attention in recent years [5], [11], [16], =-=[20]-=-, [22], [25], [27]. An (n X p)-dimensional biquadratic tensor A has the form A = (aijki), where i,j = l,...,n; k,l=l,...,p; 2 < n < p, with symmetric property ajj^i = aju^i = üijik for any i, j , k, a... |

1 |
The cubic spherical optimization problem. Math. Comp., to appear. Copyright of SIAM Journal on Matrix Analysis & Applications is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites or
- QI, YE
(Show Context)
Citation Context ...etric matrix space, we have m = 2. It is not difficult to see that App(Sym2 (91") ) = - ! = •\/n Again, it is an open question to find the exact values of App(Sym'"(9l")) for m > 3. By Theorem 2.2 of =-=[28]-=-, we have the following theorem. THEOREM 4.1. Eor any A G Sym^(9l"), we have p{A) = a{A). CoN.iECTURE 1. Eor any A G Sym'"(9l") with m>4, we still have p{A) = (J{A). PROPOSITION 4.2. max a;, ¿=i j -.x... |