Given f:{β1,1}nβ{β1,1}, define the \emph{spectral
distribution} of f to be the distribution on subsets of [n] in which the
set S is sampled with probability fβ(S)2. Then the Fourier
Entropy-Influence (FEI) conjecture of Friedgut and Kalai (1996) states that
there is some absolute constant C such that H[fβ2]β€Cβ Inf[f]. Here, H[fβ2]
denotes the Shannon entropy of f's spectral distribution, and
Inf[f] is the total influence of f. This conjecture is one
of the major open problems in the analysis of Boolean functions, and settling
it would have several interesting consequences.
Previous results on the FEI conjecture have been largely through direct
calculation. In this paper we study a natural interpretation of the conjecture,
which states that there exists a communication protocol which, given subset S
of [n] distributed as fβ2, can communicate the value of S using
at most Cβ Inf[f] bits in expectation.
Using this interpretation, we are able show the following results:
1. First, if f is computable by a read-k decision tree, then
H[fβ2]β€9kβ Inf[f].
2. Next, if f has Inf[f]β₯1 and is computable by a
decision tree with expected depth d, then H[fβ2]β€12dβ Inf[f].
3. Finally, we give a new proof of the main theorem of O'Donnell and Tan
(ICALP 2013), i.e. that their FEI+ conjecture composes.
In addition, we show that natural improvements to our decision tree results
would be sufficient to prove the FEI conjecture in its entirety. We believe
that our methods give more illuminating proofs than previous results about the
FEI conjecture