Fourier Entropy-Influence Conjecture for Random Linear Threshold Functions

Abstract

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function $f:\{+1,-1\}^n \rightarrow \{+1,-1\}$, the Fourier entropy of f is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a “random” linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on $[-1,1]$ and Normal distribution.

N. Saurabh—The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 616787.

A Reducing Non-homogeneous to Homogeneous case

Let $f(x) = \mathsf {sign}(w_{0} + \sum _{i=1}^n w_ix_i)$ for all $x \in \{+1,-1\}^n$. Recall from Eq. (2) we have an exact expression for the i-th influence for all $1 \leqslant i \leqslant n$. We can relax the probability estimate to lower bound the influence as follows,

$$\begin{aligned} \mathsf {Inf}_i(f) \geqslant \frac{1}{2}\Pr _{x \in \{+1,-1\}^n}\left[ \left| w_0 + \sum _{j \leqslant n :j \ne i} w_jx_j\right| \leqslant |w_i| \right] . \end{aligned}$$

(8)

Now consider the function $g(x_0, x_1, \dots , x_n) = \mathsf {sign}(\sum _{i=0}^{n} w_ix_i)$ by adding the extra variable $x_{0}$. We claim that $\mathsf {Inf}_i(g)\leqslant 2 \mathsf {Inf}_i(f)$, for all $1\leqslant i \leqslant n$. Fix an $i \in [n]$. From Eq. (2) we know that $\mathsf {Inf}_i(g)$ equals

$$\begin{aligned} \frac{1}{2}\left( \Pr \left[ - |w_i|< w_0 + \sum _{1 \leqslant j \leqslant n :j \ne i} w_jx_j \leqslant |w_i| \right] + \Pr \left[ -|w_i| < - w_0 +\sum _{1 \leqslant j \leqslant n :j \ne i} w_jx_j \leqslant |w_i| \right] \right) , \end{aligned}$$

where the probabilities are uniform distribution over $x_1, \dots , x_n \in \{+1,-1\}^n$. By relaxing the event in each case we have

$$\begin{aligned} \mathsf {Inf}_i(g)&\leqslant \frac{1}{2}\left( \Pr \left[ - |w_i| \leqslant w_0 + \sum _{1 \leqslant j \leqslant n :j \ne i} w_jx_j \leqslant |w_i| \right] + \Pr \left[ -|w_i| \leqslant - w_0 +\sum _{1 \leqslant j \leqslant n :j \ne i} w_jx_j \leqslant |w_i| \right] \right) . \end{aligned}$$

It is easily seen that,

$$\begin{aligned} \Pr _{x \in \{+1,-1\}^{n}}\left[ -|w_i| \leqslant w_0+\sum _{1\leqslant j \leqslant n :j \ne i} w_jx_j \leqslant |w_i| \right] = \Pr _{x \in \{+1,-1\}^{n}}\left[ -|w_i| \leqslant -w_0 + \sum _{1\leqslant j \leqslant n :j \ne i} w_jx_j \leqslant |w_i| \right] . \end{aligned}$$

Indeed, there exists a 1-1 correspondence between n-bit strings satisfying the left hand side event and the right hand side event. Thus,

$$\begin{aligned} \mathsf {Inf}_i(g) \leqslant \Pr _{x \in \{+1,-1\}^{n}}\left[ -|w_i| \leqslant w_0+\sum _{1\leqslant j \leqslant n :j \ne i} w_jx_j \leqslant |w_i| \right] \leqslant 2\cdot \mathsf {Inf}_i(f) , \end{aligned}$$

where the second inequality follows from (8).

Therefore, we have

$$\begin{aligned} \mathsf {Inf}(f) \geqslant \sum _{i=1}^{n}\frac{\mathsf {Inf}_i(g)}{2} = \frac{\mathsf {Inf}(g) - \mathsf {Inf}_0(g)}{2} \geqslant \frac{\mathsf {Inf}(g) -1}{2}. \end{aligned}$$

Thus we can translate a lower bound on influences in the homogeneous case to the non-homogeneous case.