I was wondering whether or not there's an intuitive way of understanding the Gaussian isoperimetric inequality. I have been studying the Classical isoperimetric inequality and I finally understand it. I want to move to advance isoperimetric inequalities. I am interested in the Gaussian isoperimetric as it seems to have nice and practical applications in information theory.

I have no background in measure theory, but I understand that the concept of measure is a generalization of the notions of length, area and volume. I also understand that the Gaussian measure is a probability measure, meaning that it has the additional property of being normalized.

I've also looked at the definition of half spaces. I understand what a half space is. Most resources I've found do not explain the intuition behind the inequality, like the Wikipedia page , they simply provide the definition which is not easy not to understand.

How do you interpret the inequality?


The screenshots below come from the book Mathematical Foundations of Infinite-Dimensional Statistical Models.

The classical isoperimetric inequality is well-known: given a fixed perimeter, a circle achieves the largest area, or, given a fixed area, a circle achieves the smallest perimeter among other shapes. If we recall that perimeter can be seen as the derivative of the area, this is like saying

$$ \mu(C+\epsilon O_2)\le\mu(A+\epsilon O_2) $$ where $\mu$ is Lebesgue measure, $A$ is a measurable set, $C$ is a circle with $\mu(C)=\mu(A)$, $O_2$ is the 2D unit disk, and the "$+$" is Minkowski addition. To see why, one can subtract $\mu(C)=\mu(A)$ from both two sides, divide them by $\epsilon$, and let $\epsilon\to0^+$, then he/she will get the usual form of isoperimetric inequality.

It turns out that this form of isoperimetric inequality is more convenient to generalize: we can allow higher dimensions, Riemannian manifolds equipped with some geodesic distances other than $\mathbb{R}^2$, or other measures. For example, we can have

enter image description here

where $A_\epsilon\triangleq A+\epsilon O_n$ and same for $C_\epsilon$.

We can interpret isoperimetric inequalities from the perspective of concentration inequality: it answers that if you perturb a set with some ϵ in distance/metric, how large its size will (at least) change. In the context of probabilistic measures, the change of "size" becomes a probability.

We need the following lemma to have a better understanding of or to prove Gaussian isoperimetric inequality:

enter image description here

where $\gamma_n$ is the standard Gaussian measure on $\mathbb{R}^n$. One can verify the claim via doing simulation for, say $n=1$, by projecting points uniformly distributed on $\sqrt{m}S^{m+1}$ onto $\mathbb{R}^1$. If in Python:

import numpy as np
import matplotlib.pyplot as plt

def runif_s(n_samples, n, m):
    rnorm = np.random.randn(n_samples, n + m + 1)
    return np.sqrt(m) * rnorm / np.linalg.norm(rnorm, axis=1)[..., np.newaxis]

def proj_hist(data, **kwargs):
    n = 1
    plt.hist(data[:, :n], density=True, **kwargs)
    plt.title('m = %d' % (data.shape[1] - n - 1))

if __name__ == '__main__':
    n = 1
    n_figures = 5
    n_samples = 10**4
    [proj_hist(runif_s(n_samples, n, int(m)), bins='auto') for m in np.logspace(0, 2, n_figures)]

And you may see the projected distribution is visually close to the normal distribution when $m$ grows large: enter image description here

Let's return to Gaussian isoperimetric inequality. The inequality is a version of isoperimetric inequality w.r.t. Gaussian measure, by which we roughly mean, finding the counterpart of a "circle" under Gaussian measure. Recall that we already have an isoperimetrc inequality for $S^{m+n}$ w.r.t. Lebesgue measure (Theorem 2.2.1), and a relation between this measure and the standard Gaussian measure on $\mathbb{R}^n$ (Lemma 2.2.2), so all we have to do is to project the cap on $S^{m+n}$ back to $\mathbb{R}^n$, and let $m\to\infty$. For the convenience of doing projection, we choose the cap "perpendicular" to $\mathbb{R}^n$. If we look into the case of $n=1$ to gain intuition, the cap will be symmetric around $\mathbb{R}^1$ with the pole lying at $-\sqrt{m}$.

Let's say we now have a measurable set $A$ on $\mathbb{R}^1$, then we can find the cap $C$ on $S^{m+1}$ with $\mu(C)=\gamma_1(A)$. The projection of the cap onto $\mathbb{R}^1$ is the interval $[-\sqrt{m},b(m)]$ for some $b(m)$. By taking $m\to\infty$, the interval becomes $(-\infty,b(\infty)]$, and according to Lemma 2.2.2, we know that $b(\infty)=\Phi^{-1}(\gamma_1(A))$, giving the "circle" w.r.t. $\gamma_1$ being $\{x:x\le \Phi^{-1}(\gamma_1(A))\}$.

The proof of the general Gaussian isoperimetric inequality follows the same intuition, except that we need to replace the ray $\{x:x\le \Phi^{-1}(\gamma_1(A))\}$ with the hyperplane $\{x:\langle x,u \rangle\le\Phi^{-1}(\gamma_n(A))\}$, where $u$ is an arbitrary unit vector in $\mathbb{R}^n$. Finally we will have

enter image description here

and its countably-infinite-dimensional version

enter image description here where $\mathcal{C}$ is the cylindrical $\sigma$-algebra on $\mathbb{R}^\mathbb{N}$.

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