On the Fourier Entropy Influence Conjecture for Extremal Classes

The Fourier Entropy-Influence (FEI) Conjecture of Friedgut and Kalai states that ${\bf H}[f] \leq C \cdot {\bf I}[f]$ holds for every Boolean function $f$, where ${\bf H}[f]$ denotes the spectral entropy of $f$, ${\bf I}[f]$ is its total influence, and $C > 0$ is a universal constant. Despite significant interest in the conjecture it has only been shown to hold for some classes of Boolean functions such as symmetric functions and read-once formulas. In this work, we prove the conjecture for extremal cases, functions with small influence and functions with high entropy. Specifically, we show that: * FEI holds for the class of functions with ${\bf I}[f] \leq 2^{-cn}$ with the constant $C = 4 \cdot \frac{c+1}{c}$. Furthermore, proving FEI for a class of functions with ${\bf I}[f] \leq 2^{-s(n)}$ for some $s(n) = o(n)$ will imply FEI for the class of all Boolean functions. * FEI holds for the class of functions with ${\bf H}[f] \geq cn$ with the constant $C = \frac{1 + c}{h^{-1}(c^2)}$. Furthermore, proving FEI for a class of functions with ${\bf H}[f] \geq s(n)$ for some $s(n) = o(n)$ will imply FEI for the class of all Boolean functions. Additionally, we show that FEI holds for the class of functions with constant $\|\widehat{f}\|_1$, completing the results of Chakhraborty et al. that bounded the entropy of such functions. We also improve the result of Wan et al. for read-k decision trees, from ${\bf H}[f] \leq O(k) \cdot {\bf I}[f]$ to ${\bf H}[f] \leq O(\sqrt{k}) \cdot {\bf I}[f]$. Finally, we suggest a direction for proving FEI for read-k DNFs, and prove the Fourier Min-Entropy/Influence (FMEI) Conjecture for regular read-k DNFs.