219 Boolean Analysis — Bobkov
219.1 Overview
Bobkov’s functional measures the expected Euclidean length of a Gaussian profile together with the discrete gradient. The cube inequality bounds the profile of the mean, and the functional is monotone in the Boolean noise correlation.
219.2 Declarations
For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). The Bobkov functional assigns the value \(B(f)\) to \(f\).
Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For real \(a,b\) with \(a+b,a-b\in [0,1]\), \(\overline I(a)\le \bigl(\sqrt{\overline I(a+b)^2+b^2}+\sqrt{\overline I(a-b)^2+b^2}\bigr)/2\).
If \(f:\{ 0,1\} ^n\to [0,1]\), then its uniform mean satisfies \(\mathbb E f\in [0,1]\).
For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). In dimension \(n=0\), \(B(f)=\overline I(\mathbb E f)\).
Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(f:\{ 0,1\} ^{n+1}\to [0,1]\), put \(f_b(x)=f(x,b)\) for \(b\in \{ 0,1\} \), and set \(d=(f_0-f_1)/2\). Then \(\overline I(\mathbb E f)\le \bigl(\sqrt{\overline I(\mathbb E f_0)^2+(\mathbb E d)^2}+\sqrt{\overline I(\mathbb E f_1)^2+(\mathbb E d)^2}\bigr)/2\).
For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). If \(d\in \mathbb R\) and \(\overline I(\mathbb E f)\le B(f)\), then \(\sqrt{\overline I(\mathbb E f)^2+d^2}\le \sqrt{B(f)^2+d^2}\).
Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), \(f_b(x)=f(x,b)\), and \(d=(f_0-f_1)/2\). For every \(b\in \{ 0,1\} \) and \(x\in \{ 0,1\} ^n\), \(\sqrt{(\sqrt{\overline I(f_b(x))^2+\sum _{i\lt n}(D_i f_b(x))^2})^2+d(x)^2}=\sqrt{\overline I(f(x,b))^2+\sum _{i\lt n+1}(D_i f(x,b))^2}\), where \(D_i\) denotes the coordinate half-difference.
For a real function \(h\) on a uniform Boolean cube, put \(B(h)=\mathbb E\sqrt{\overline I(h)^2+\sum _i(D_i h)^2}\), where \(D_i\) is the coordinate half-difference and \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\) and \(d=(f_0-f_1)/2\). If \(\overline I(\mathbb E f_b)\le B(f_b)\), then \(\sqrt{\overline I(\mathbb E f_b)^2+(\mathbb E d)^2}\le \mathbb E_x\sqrt{\overline I(f(x,b))^2+\sum _{i\lt n+1}(D_i f(x,b))^2}\).
For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For every \(n\ge 0\) and \(f:\{ 0,1\} ^n\to [0,1]\), \(\overline I(\mathbb E f)\le B(f)\).
For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For \(f:\{ 0,1\} ^n\to [0,1]\) and \(0\le \rho \le 1\), \(B(T_\rho f)\le B(f)\), where \(T_\rho \) is the Boolean noise operator with correlation \(\rho \).
For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For \(f:\{ 0,1\} ^n\to [0,1]\), the map \(\rho \mapsto B(T_\rho f)\) is increasing on \([0,1]\), where \(T_\rho \) is Boolean noise with correlation \(\rho \).