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

Stegeman, Alwin; Sidiropoulos, Nicholas D.

doi:10.1016/j.laa.2006.08.010

Cited by 202 publications

(152 citation statements)

References 10 publications

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“…The Kruskal-rank of , denoted , is the maximum such that any columns of are linearly independent . Given , if , then are unique up to a common column permutation and scaling, i.e., , , , where is a permutation matrix and non-singular diagonal matrices such that , see [5]- [8], [13]- [15].…”

Section: Tensor Decomposition Preliminariesmentioning

confidence: 99%

Multi-Way Compressed Sensing for Sparse Low-Rank Tensors

Sidiropoulos

Kyrillidis

2012

IEEE Signal Process. Lett.

124

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Abstract-For linear models, compressed sensing theory and methods enable recovery of sparse signals of interest from few measurements-in the order of the number of nonzero entries as opposed to the length of the signal of interest. Results of similar flavor have more recently emerged for bilinear models, but no results are available for multilinear models of tensor data. In this contribution, we consider compressed sensing for sparse and low-rank tensors. More specifically, we consider low-rank tensors synthesized as sums of outer products of sparse loading vectors, and a special class of linear dimensionality-reducing transformations that reduce each mode individually. We prove interesting "oracle" properties showing that it is possible to identify the uncompressed sparse loadings directly from the compressed tensor data. The proofs naturally suggest a two-step recovery process: fitting a low-rank model in compressed domain, followed by per-mode decompression. This two-step process is also appealing from a computational complexity and memory capacity point of view, especially for big tensor datasets.Index Terms-, CANDECOMP/PARAFAC, compressed sensing, multi-way analysis, tensor decomposition.

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Section: Tensor Decomposition Preliminariesmentioning

confidence: 99%

Multi-Way Compressed Sensing for Sparse Low-Rank Tensors

Sidiropoulos

Kyrillidis

2012

IEEE Signal Process. Lett.

124

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“…The following theorem establishes a condition under which essential uniqueness is guaranteed [39], [40], [53], [59].…”

Section: Definitionmentioning

confidence: 99%

Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization

Lathauwer

Castaing

2008

IEEE Trans. Signal Process.

188

134

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Abstract-In this paper, we study simultaneous matrix diagonalization-based techniques for the estimation of the mixing matrix in underdetermined independent component analysis (ICA). This includes a generalization to underdetermined mixtures of the well-known SOBI algorithm. The problem is reformulated in terms of the parallel factor decomposition (PARAFAC) of a higher-order tensor. We present conditions under which the mixing matrix is unique and discuss several algorithms for its computation.Index Terms-Canonical decomposition, higher order tensor, independent component analysis (ICA), parallel factor (PARAFAC) analysis, simultaneous diagonalization, underdetermined mixture.

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“…Parameter identifiability of the model with DAG shown in Figure 2 is an instance of a more general theorem of Kruskal [13] (see also [18,19]). However, known proofs of the full Kruskal theorem do not yield an explicit procedure for recovering parameters.…”

Section: Two Special Modelsmentioning

confidence: 99%

Parameter Identifiability of Discrete Bayesian Networks with Hidden Variables

Allman

Rhodes

Stanghellini

et al. 2015

Journal of Causal Inference

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Identifiability of parameters is an essential property for a statistical model to be useful in most settings. However, establishing parameter identifiability for Bayesian networks with hidden variables remains challenging. In the context of finite state spaces, we give algebraic arguments establishing identifiability of some special models on small directed acyclic graphs (DAGs). We also establish that, for fixed state spaces, generic identifiability of parameters depends only on the Markov equivalence class of the DAG. To illustrate the use of these results, we investigate identifiability for all binary Bayesian networks with up to five variables, one of which is hidden and parental to all observable ones. Surprisingly, some of these models have parameterizations that are generically 4-to-one, and not 2-to-one as label swapping of the hidden states would suggest. This leads to interesting conflict in interpreting causal effects.

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

Cited by 202 publications

References 10 publications

Multi-Way Compressed Sensing for Sparse Low-Rank Tensors

Multi-Way Compressed Sensing for Sparse Low-Rank Tensors

Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization

Parameter Identifiability of Discrete Bayesian Networks with Hidden Variables

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