Dimension-free exponential concentration for log-concave measures

Determine whether every log-concave probability measure ν on R^d with convex potential W and covariance operator norm at most 1 satisfies an exponential concentration inequality of the form ν(f ≥ ∫ f dν + r) ≤ c exp(−r/σ) for all 1-Lipschitz functions f and all r ≥ 0 with universal constants c and σ independent of the dimension d, i.e., remove the √(log d) factor appearing in the best known bound for such measures.

Background

The paper derives growth estimates for Brenier optimal transport maps using concentration inequalities of the target measure. For log-concave targets, the authors invoke a known exponential concentration result that currently incurs a √(log d) loss in the concentration scale parameter.

They note that it is conjectured this dimensional factor is unnecessary, which, if resolved affirmatively, would yield dimension-free exponential concentration for isotropic (or appropriately normalized) log-concave measures and would sharpen the transport growth bounds established in this work.