Sharper bounds improve understanding of entropy and stability in math
Hermite-Fisher bounds and stability for min-entropy power inequalities
Information Theory
Summary
This paper finds better ways to measure differences between certain types of random data using special math functions called Hermite polynomials. The authors also improve important math inequalities related to uncertainty and randomness, which apply to multiple dimensions. Along the way, they prove a new stability result for a geometric problem involving shapes called Euclidean balls. These improvements help better understand how randomness behaves and how tightly certain mathematical bounds hold.
What this means in practice
- •For information theorists: Use sharper lower bounds on information measures to analyze data compression and transmission limits more precisely.
- •For geometric analysts: Apply newly established stability results to better understand geometric inequalities involving high-dimensional shapes.
A theory result. No direct application yet.
Authors
Silouanos Brazitikos, Martin Rapaport, Tomasz Tkocz
Abstract
We derive explicit lower bounds for relative Fisher information by combining a variational principle with suitably orthogonalized Hermite-polynomial test functions. The resulting cumulant bounds are asymptotically sharp and yield lower bounds for Gaussian entropy deficits. We also establish quantitative versions of sharp min-entropy power inequalities in all dimensions. En route, we develop a stability result for Brzezinski's sharp bound for block sections of products of Euclidean balls, which may be of independent interest.