Given a tensor $f$ in a Euclidean tensor space, we are interested in the critical points of the distance function from $f$ to the set of tensors of rank at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When $f$ is a matrix, the critical rank-one matrices for $f$ correspond to the singular pairs of $f$. The critical rank-one tensors for $f$ lie in a linear subspace $H_f$, the critical space of $f$. Our main result is that, for any $k$, the critical rank-at-most-$k$ tensors for a sufficiently general $f$ also lie in the critical space $H_f$. This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space $H_f$ is spanned by the complex critical rank-one tensors. Since $f$ itself belongs to $H_f$, we deduce that also $f$ itself is a linear combination of its critical rank-one tensors.

Best rank-k approximations for tensors: generalizing Eckart–Young / Jan Draisma, Giorgio Ottaviani, Alicia Tocino. - In: RESEARCH IN THE MATHEMATICAL SCIENCES. - ISSN 2197-9847. - ELETTRONICO. - 5:(2018), pp. 0-0. [10.1007/s40687-018-0145-1]

Best rank-k approximations for tensors: generalizing Eckart–Young

Giorgio Ottaviani;Alicia Tocino
2018

Abstract

Given a tensor $f$ in a Euclidean tensor space, we are interested in the critical points of the distance function from $f$ to the set of tensors of rank at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When $f$ is a matrix, the critical rank-one matrices for $f$ correspond to the singular pairs of $f$. The critical rank-one tensors for $f$ lie in a linear subspace $H_f$, the critical space of $f$. Our main result is that, for any $k$, the critical rank-at-most-$k$ tensors for a sufficiently general $f$ also lie in the critical space $H_f$. This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space $H_f$ is spanned by the complex critical rank-one tensors. Since $f$ itself belongs to $H_f$, we deduce that also $f$ itself is a linear combination of its critical rank-one tensors.
2018
5
0
0
Jan Draisma, Giorgio Ottaviani, Alicia Tocino
File in questo prodotto:
File Dimensione Formato  
Draisma_et_al-2018-Research_in_the_Mathematical_Sciences.pdf

accesso aperto

Descrizione: Articolo principale
Tipologia: Pdf editoriale (Version of record)
Licenza: Open Access
Dimensione 527.98 kB
Formato Adobe PDF
527.98 kB Adobe PDF

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/2158/1136807
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 11
  • ???jsp.display-item.citation.isi??? 10
social impact