On the uniqueness of multilinear decomposition of N‐way arrays

By N. Sidiropoulos, Rasmus Bro. 2000

606 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Computer Networks and Communications

View Paper: https://www.semanticscholar.org/paper/bced963ba7966b0566bafa909914b857079cb816

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In Computational Mathematics

In Computational Mechanics

In Computer Networks and Communications

Citations in the corpus

Hyperspectral image noise reduction based on rank-1 tensor decomposition (2013, background): “In order to achieve a unique solution for the factor matrices, it is assumed that the columns of U, V, and W are normalized to length one (Sidiropoulos and Bro, 2000), i.”
Accelerated Canonical Polyadic Decomposition Using Mode Reduction (2012, neutral): “…in 1977 [18] for 3-way tensors and then extended for N th-order tensors by Sidiropoulos and Bro in 2000 [19]: Lemma 2 (Uniqueness condition for CPD [19]): For an N th-order tensor Y = A (1) , A (2) then the decomposition is essentially unique, where kr A ( n ) is the Kruskal rank of A ( n ) , n ∈…”
Tensor decompositions, alternating least squares and other tales (2009, background): “The proof has been later simpliﬁed [55,56], and extended to tensors of order d higherthan3[57].”
Tensors for Data Mining and Data Fusion (2016, background): “In particular, it is shown that CP is unique, under very mild conditions, up to scaling and permutation of the R components [Kruskal 1977; Sidiropoulos and Bro 2000; ten Berge and Sidiropoulos 2002; Jiang and Sidiropoulos 2004; Stegeman et al. 2006; Chiantini and Ottaviani 2012].”
A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization (2006, positive): “The result was generalized to tensors of arbitrary order in [ 40 ].”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, background): “Figure 4 illustrates the concept of MWCA and its ﬂexibility in choosing the mode-wise constraints; a Tucker representation of MWCA naturally accommodates such diversities in different modes.”
On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode (2009, methodology): “A proof of Kruskal’s uniqueness condition for complex-valued CP can be found in Sidiropoulos and Bro [13].”
Tensors and Hypermatrices (2013, background): “[SB00] Fact 8 has been generalized to arbitrary order d ≥ 3.”
Partial uniqueness in CANDECOMP/PARAFAC (2004, background): “Sidiropoulos and Bro [6] have given a shortcut proof for Kruskal’s sufficient condition and generalized it to n-way arrays (n> 3).”
Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects (2015, background): “This is analogous to the previously-mentioned result (Section III-C2) for a single tensor, that increasing its order N relaxes the bound on R [57], [91].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, positive): “Under rather mild conditions, the CP decomposition is generally unique by itself [237], and does not require additional constraints on the factor matrices to achieve uniqueness, which makes it a powerful and useful tool.”
Non-negative tensor factorization with applications to statistics and computer vision (2005, background): “…of tensor ranks, is that unlike matrix factorization, which is generally non-unique for any rank greater than one, a 3-valence tensor decomposition is essentially unique under mild conditions (Kruksal, 1977) and the situation actually improves in higher dimensions n > 3 (Sidiropoulos & Bro, 2000).”
CANDECOMP/PARAFAC Decomposition of High-Order Tensors Through Tensor Reshaping (2012, methodology): “By exploiting the uniqueness of CPD under mild conditions (see discussion on uniqueness conditions for order-3 CPD in [33], [34], [27] and for higher order CPD in [35]–[37]), and the relation of a tensor in the Kruskal form [38] and its unfolded tensor, we develop a fast approach for high order and ”
On generic identifiability of symmetric tensors of subgeneric rank (2015, background): “A popular criterion for testing identifiability that is applicable for Veronese embeddings of any degree d ≥ 3 is the so-called Kruskal condition [24,33].”