In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-1 terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, but few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semialgebraic set enclosing the set of rank-r tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for symmetric tensors of small dimension as well as for binary forms of degree at least three. We extend these results to 4 x 4 X 4 X 4 symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank 8, which is optimal. © 2017 Society for Industrial and Applied Mathematics.

Effective criteria for specific identifiability of tensors and forms / Chiantini, Luca; Ottaviani, GIORGIO MARIA; Vannieuwenhoven, NICK JOS. - In: SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. - ISSN 0895-4798. - STAMPA. - 38:(2017), pp. 656-681. [10.1137/16M1090132]

Effective criteria for specific identifiability of tensors and forms

Ottaviani, Giorgio;VANNIEUWENHOVEN, NICK JOS
2017

Abstract

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-1 terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, but few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semialgebraic set enclosing the set of rank-r tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for symmetric tensors of small dimension as well as for binary forms of degree at least three. We extend these results to 4 x 4 X 4 X 4 symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank 8, which is optimal. © 2017 Society for Industrial and Applied Mathematics.
2017
38
656
681
Chiantini, Luca; Ottaviani, GIORGIO MARIA; Vannieuwenhoven, NICK JOS
File in questo prodotto:
File Dimensione Formato  
cov3_20160507.pdf

accesso aperto

Tipologia: Versione finale referata (Postprint, Accepted manuscript)
Licenza: Creative commons
Dimensione 510.33 kB
Formato Adobe PDF
510.33 kB Adobe PDF
16m1090132.pdf

Accesso chiuso

Tipologia: Pdf editoriale (Version of record)
Licenza: Tutti i diritti riservati
Dimensione 542.54 kB
Formato Adobe PDF
542.54 kB Adobe PDF   Richiedi una copia

I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1103187
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 49
  • ???jsp.display-item.citation.isi??? 44
social impact