A comparison of different notions of ranks of symmetric tensors

By Alessandra Bernardi, Jérôme Brachat, Bernard Mourrain. 2012

49 citations recorded by the source corpus.

Abstract We introduce various notions of rank for a high order symmetric tensor taking values over the complex numbers, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.

Subfields: Computational Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/000bc16b749fc9c1af108b13e7b88dc70f5c162b

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In Computational Mathematics

1. Alessandra Bernardi
2. A. Oneto
3. Alessandro Gimigliano
4. Enrico Carlini
5. M. Catalisano

Citations in the corpus

• The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “We refer to [82] for a complete analysis on the relations between different notions of ranks.”