On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

Abstract

The Schmidt-Eckart-Young theorem for matrices states that the optimal rank-r approximation to a matrix is obtained by retaining the first r terms from the singular value decomposition of that matrix. This work considers a generalization of this optimal truncation property to the CANDECOMP/PARAFAC decomposition of tensors and establishes a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure in complex tensor spaces. It is proved moreover that for complex tensors of small rank this condition can only be satisfied by a set of tensors of Lebesgue measure zero.Keywords : Canonical Polyadic decomposition, CANDECOMP, PARAFAC, orthogonal rank decomposition, Tensor singular value decomposition, EckartYoung theorem MSC : Primary : 14A10, 14A25, 14Q15, 15A21, 15A69, 15B99. ON GENERIC NONEXISTENCE OF THE SCHMIDT-ECKART-YOUNG DECOMPOSITION FOR COMPLEX TENSORSN. VANNIEUWENHOVEN † , J. NICAISE ‡ , R. VANDEBRIL † , AND K. MEERBERGEN † Abstract. The Schmidt-Eckart-Young theorem for matrices states that the optimal rank-r approximation to a matrix is obtained by retaining the first r terms from the singular value decomposition of that matrix. This work considers a generalization of this optimal truncation property to the CANDECOMP/PARAFAC decomposition of tensors and establishes a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure in complex tensor spaces. It is proved moreover that for complex tensors of small rank this condition can only be satisfied by a set of tensors of Lebesgue measure zero.

An Algorithm For Generic and Low-Rank Specific Identifiability of Complex Tensors

“…This would mean that it takes no more than about two weeks per experiment. In fact, in [50], it was verified with the second computer system using a modified version of the presented algorithm that (PC 14 ) 4 , where Downloaded 11/20/14 to 108.179.154.91. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Table 1 A comparison between the maximum rank for which generic r-identifiability can be proved for S = PC m × PC n × PC n using the Domanov [10], which also proves generic identifiability, requires approximately 10 minutes for proving generic r-identifiability of (PC 5 ) 4 , where Π = 625, on the second computer system using one processing core and Macaulay2 v1.4; with the method in this paper, the same result was proved in less than one second.…”

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

“…However, whether nonexistence of a best rank-R approximation holds on a set of positive volume is still an open problem. A related result by [42] concerns tensors that allow a Schmidt-Eckart-Young (SEY) decomposition: a decomposition into a sum of rank-1 tensors with a best rank-R approximation given by the sum of R of these rank-1 tensors. For matrices the SEY decomposition is given by the SVD [43].…”

“…For matrices the SEY decomposition is given by the SVD [43]. The result of [42] is that the set of complex order-N tensors that do not admit an SEY decomposition has positive volume for N ≥ 3. Since the set of complex tensors that do not have a best rank-R approximation is a subset of the set of complex tensors that do not admit an SEY decomposition, this does not imply that the former set also has positive volume.…”

Parallel tensor methods for high-dimensional linear PDEs

“…2 A diagonalization of the core tensor in (14) would minimize the number of terms in the series expansion. Unfortunately, this is impossible for a tensor with dimension larger than 2 (see, e.g., [47,48,43,49]) and, for complex tensors, [50]). A closer look at the canonical tensor decomposition (1) reveals that such an expansion is in the form of a fully diagonal high-order Schmidt decomposition, i.e.,…”