Large deviations, moderate deviations, and the KLS conjecture

2020/03/25 by Alonso-Gutiérrez, David, Prochno, Joscha, Thaele, Christoph
#46B09 #52A23 Secondary: 60D05 #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 60F10 #Probability (math.PR)

paper · doi:10.48550/arxiv.2003.11442

Abstract

Having its origin in theoretical computer science, the Kannan-Lovász-Simonovits (KLS) conjecture is one of the major open problems in asymptotic convex geometry and high-dimensional probability theory today. In this work, we establish a new connection between this conjecture and the study of large and moderate deviations for isotropic log-concave random vectors, thereby providing a novel possibility to tackle the conjecture. We then study the moderate deviations for the Euclidean norm of random orthogonally projected random vectors in an ℓ_pⁿ-ball. This leads to a number of interesting observations: (A) the ℓ₁ⁿ-ball is critical for the new approach; (B) for p≥ 2 the rate function in the moderate deviations principle undergoes a phase transition, depending on whether the scaling is below the square-root of the subspace dimensions or comparable; (C) for 1≤ p<2 and comparable subspace dimensions, the rate function again displays a phase transition depending on its growth relative to n^p/2.

Related

• The large and moderate deviations approach in geometric functional analysis
• Large deviations for high-dimensional random projections of ℓ_pⁿ-balls
• Large deviations for random projections of ℓ^p balls
• High-dimensional limit theorems for random vectors in ℓ_pⁿ-balls. II
• High-dimensional limit theorems for random vectors in ℓ_pⁿ-balls