What is the probability that a random symmetric tensor is close to rank-one?

Fuente: arXiv

Saved in:

Bibliographic Details
Main Authors:	Cazzaniga, Alberto, Lerario, Antonio, Rosana, Andrea
Format:	Preprint
Published:	2023
Subjects:	Algebraic Geometry Differential Geometry Probability
Online Access:	Acceder al recurso
Tags:	Add Tag No Tags, Be the first to tag this record!

_version_	1866929620076462080
author	Cazzaniga, Alberto Lerario, Antonio Rosana, Andrea
author_facet	Cazzaniga, Alberto Lerario, Antonio Rosana, Andrea
contents	We address the general problem of estimating the probability that a real symmetric tensor is close to rank-one tensors. Using Weyl's tube formula, we turn this question into a differential geometric one involving the study of metric invariants of the real Veronese variety. More precisely, we give an explicit formula for its reach and curvature coefficients with respect to the Bombieri-Weyl metric. These results are obtained using techniques from Random Matrix theory and an explicit description of the second fundamental form of the Veronese variety in terms of GOE matrices. Our findings give a complete solution to the original problem. In the case of rational normal curves it leads to a simple formula describing explicitly exponential decay with respect to the degree of the tensor.
format	Preprint
id	arxiv_https___arxiv_org_abs_2301_05502
institution	arXiv
publishDate	2023
record_format	arxiv
spellingShingle	What is the probability that a random symmetric tensor is close to rank-one? Cazzaniga, Alberto Lerario, Antonio Rosana, Andrea Algebraic Geometry Differential Geometry Probability We address the general problem of estimating the probability that a real symmetric tensor is close to rank-one tensors. Using Weyl's tube formula, we turn this question into a differential geometric one involving the study of metric invariants of the real Veronese variety. More precisely, we give an explicit formula for its reach and curvature coefficients with respect to the Bombieri-Weyl metric. These results are obtained using techniques from Random Matrix theory and an explicit description of the second fundamental form of the Veronese variety in terms of GOE matrices. Our findings give a complete solution to the original problem. In the case of rational normal curves it leads to a simple formula describing explicitly exponential decay with respect to the degree of the tensor.
title	What is the probability that a random symmetric tensor is close to rank-one?
topic	Algebraic Geometry Differential Geometry Probability
url	https://arxiv.org/abs/2301.05502