2018/06/10 by Guy Shalev, Shalev, Guy · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.1806.03646
openalex publication_date 2018/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Fourier Entropy-Influence (FEI) Conjecture of Friedgut and Kalai states that \bf H[f] ≤ C ⋅ \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] ≤ 2-cn with the constant C = 4 ⋅ (c+1)/(c). Furthermore, proving FEI for a class of functions with \bf I[f] ≤ 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] ≥ cn with the constant C = \frac1 + ch-1(c2). Furthermore, proving FEI for a class of functions with \bf H[f] ≥ 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 ‖\widehatf‖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] ≤ O(k) ⋅ \bf I[f] to \bf H[f] ≤ O(√(k)) ⋅ \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.