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Article Dans Une Revue Communications in Contemporary Mathematics Année : 2022

On the geometry of polytopes generated by heavy-tailed random vectors

Olivier Guédon
  • Fonction : Auteur
Felix Krahmer
  • Fonction : Auteur
Christian Kümmerle
  • Fonction : Auteur
Holger Rauhut
  • Fonction : Auteur

Résumé

We study the geometry of centrally symmetric random polytopes, generated by [Formula: see text] independent copies of a random vector [Formula: see text] taking values in [Formula: see text]. We show that under minimal assumptions on [Formula: see text], for [Formula: see text] and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector — namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors, we recover the estimates that were obtained previously, and thanks to the minimal assumptions on [Formula: see text], we derive estimates in cases that were out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when [Formula: see text] is [Formula: see text]-stable or when [Formula: see text] has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing — noise blind sparse recovery.

Dates et versions

hal-03939547 , version 1 (15-01-2023)

Identifiants

Citer

Olivier Guédon, Felix Krahmer, Christian Kümmerle, Shahar Mendelson, Holger Rauhut. On the geometry of polytopes generated by heavy-tailed random vectors. Communications in Contemporary Mathematics, 2022, 24 (03), ⟨10.1142/S0219199721500565⟩. ⟨hal-03939547⟩
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