On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition

By Alwin Stegeman, N. Sidiropoulos. 2007

258 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/30861b5a7876d674dd55634e890f19b6b5cf4263

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In Computational Mathematics

In Computational Mechanics

In Signal Processing

Citations in the corpus

Decomposition of homogeneous polynomials with low rank (2010, background): “For non-symmetric tensors, see Kruskal’s works and their extensions and simplifications ([17], [16], [25]).”
Tensors : A brief introduction (2014, unknown): “The most well known is that published by Kruskal [47] and extended later in [73], [81]; alternate proofs have been derived in [68], [49].”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, methodology): “The overall proof was reformulated in terms of accessible basic linear algebra in [54].”
Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects (2015, background): “, [72], [92]) where denotes the column-wise Khatri-Rao product and kA is the Kruskal-rank of matrix A, equal to the largest integer kA such that every subset of kA columns of A is linearly independent [67].”
Accelerated Canonical Polyadic Decomposition Using Mode Reduction (2012, unknown): “Before moving on, ﬁrst we assume that kr ( A ( n ) ) ≥ 2 , ∀ n ∈ N , as it is a necessary condition for uniqueness of CPD [23].”
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 ”
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].”
Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO Radar (2010, unknown): “In this sense, PARAFAC is one possible generalization of the matrix singular value decomposition (SVD) to the higher-order case.”
Subtracting a best rank-1 approximation may increase tensor rank (2009, background): “tions of tensor decompositions, refer to [19]. An important advantage of using tensor decompositions of order 3 and higher, is that the decomposition is rotationally unique under mild conditions [22] [33]. This is not the case for most matrix decompositions, e.g. Principal Component Analysis. Howeve”
On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode (2009, methodology): “A more condensed and accessible proof of (1.3) than Kruskal’s original proof was given by Stegeman and Sidiropoulos [22].”