Subtracting a best rank-1 approximation may increase tensor rank

Abstract : Is has been shown that a best rank-R approximation of an order-k tensor may not exist when R is at most 2 and k is at most 3. This poses a serious problem to data analysts using Candecomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and substracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2x2x2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2x2x2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation will result in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation has increased the tensor rank.
keyword : tenseur tensor
Document type :
Conference papers
European Signal Processing Conference, Aug 2009, Glasgow, United Kingdom. pp.25/08/2009, 2009


https://hal.archives-ouvertes.fr/hal-00435877
Contributor : Pierre Comon <>
Submitted on : Thursday, November 26, 2009 - 4:13:37 PM
Last modification on : Thursday, November 26, 2009 - 5:27:30 PM
Document(s) archivé(s) le : Thursday, June 17, 2010 - 9:56:55 PM

Files

eusStegCom12.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00435877, version 1

Collections

Citation

Alwin Stegeman, Pierre Comon. Subtracting a best rank-1 approximation may increase tensor rank. European Signal Processing Conference, Aug 2009, Glasgow, United Kingdom. pp.25/08/2009, 2009. <hal-00435877>

Export

Share

Metrics

Record views

164

Document downloads

63