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Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices
Alwin Stegeman1,2, Jos M F Ten Berge3, Lieven De Lathauwer4
1University of Groningen, Groningen. a.w.stegeman@rug.nl.
Psychometrika
|February 16, 2017
Summary
We found new conditions for the essential uniqueness of Candecomp/Parafac and Indscal decompositions in three-way arrays. These conditions are more necessary than previous ones, improving data analysis reliability.
Area of Science:
- Multivariate statistics
- Chemometrics
- Data analysis
Background:
- Candecomp/Parafac and Indscal are methods for analyzing three-way arrays.
- Essential uniqueness of the trilinear decomposition is crucial for reliable results.
- Existing uniqueness conditions may not be sufficiently necessary.
Purpose of the Study:
- To examine the uniqueness of Candecomp/Parafac and Indscal decompositions.
- To derive new, more necessary uniqueness conditions for these models.
- To provide improved theoretical underpinnings for three-way array analysis.
Main Methods:
- Analysis of three-way arrays using Candecomp/Parafac and Indscal models.
- Consideration of specific conditions for component matrices (randomly sampled, full column rank).
- Derivation of almost sure sufficient uniqueness conditions based on array order and component number.
Main Results:
- New, almost sure sufficient uniqueness conditions were obtained for both Candecomp/Parafac and Indscal.
- These conditions depend only on the order of the three-way array and the number of components.
- The derived conditions are demonstrated to be closer to necessity than Kruskal's classical condition.
Conclusions:
- The study provides refined uniqueness conditions for Candecomp/Parafac and Indscal.
- These improved conditions enhance the reliability and interpretability of trilinear decompositions.
- The findings contribute to a deeper theoretical understanding of three-way data analysis.
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