1. Concentration on the Boolean hypercube via pathwise stochastic analysis
- Author
-
Renan Gross and Ronen Eldan
- Subjects
FOS: Computer and information sciences ,Discrete mathematics ,Vertex (graph theory) ,Discrete Mathematics (cs.DM) ,General Mathematics ,Probability (math.PR) ,05 social sciences ,Boolean analysis ,Stochastic calculus ,Boundary (topology) ,02 engineering and technology ,Upper and lower bounds ,0502 economics and business ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Mathematics - Combinatorics ,050206 economic theory ,020201 artificial intelligence & image processing ,Combinatorics (math.CO) ,Isoperimetric inequality ,Boolean function ,Constant (mathematics) ,Mathematics - Probability ,Computer Science - Discrete Mathematics ,Mathematics - Abstract
We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we 1. Settle a conjecture of Talagrand [Tal97] proving that $$\int_{\left\{ -1,1\right\} ^{n}}\sqrt{h_{f}\left(x\right)}d\mu\geq C\cdot\mathrm{var}\left(f\right)\cdot\left(\log\left(\frac{1}{\sum\mathrm{Inf}_{i}^{2}\left(f\right)}\right)\right)^{1/2},$$ where $h_{f}\left(x\right)$ is the number of edges at $x$ along which $f$ changes its value, and $\mathrm{Inf}_{i}\left(f\right)$ is the influence of the $i$-th coordinate. 2. Strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function $f$, $\mathrm{var}\left(f\right)\leq C\sum_{i=1}^{n}\frac{\mathrm{Inf}_{i}\left(f\right)}{1+\log\left(1/\mathrm{Inf}_{i}\left(f\right)\right)}$. We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn-Kalai-Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. 3. Improve a quantitative relation between influences and noise stability given by Keller and Kindler. Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs., Comment: 48 pages, 2 figures
- Published
- 2022