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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

Felix Krahmer
  • Fonction : Auteur
Christian Kümmerle
  • Fonction : Auteur
Shahar Mendelson
  • Fonction : Auteur
Holger Rauhut
  • Fonction : Auteur
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Résumé

We study the geometry of centrally-symmetric random polytopes, generated by N independent copies of a random vector X taking values in R n. We show that under minimal assumptions on X, for N n and with high probability, the polytope contains a determin-istic 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 have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing-noise blind sparse recovery.
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Dates et versions

hal-02276997 , version 1 (03-09-2019)

Identifiants

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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-02276997⟩
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