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Math 401, Paper 1, Side note 3: Levy's concentration theorem
Basic definitions
Lipschitz function
$\eta$-Lipschitz function
Let (X,\operatorname{dist}_X) and (Y,\operatorname{dist}_Y) be two metric spaces. A function f:X\to Y is said to be $\eta$-Lipschitz if there exists a constant L\in \mathbb{R} such that
\operatorname{dist}_Y(f(x),f(y))\leq L\operatorname{dist}_X(x,y)
for all x,y\in X. And \eta=\|f\|_{\operatorname{Lip}}=\inf_{L\in \mathbb{R}}L.
That basically means that the function f should not change the distance between any two pairs of points in X by more than a factor of L.
Sub-Gaussian concentration
Random sampling on the CP^n
Levy's concentration theorem in High-dimensional probability by Roman Vershynin
Levy's concentration theorem (Vershynin's version)
This theorem is exactly the 5.1.4 on the High-dimensional probability by Roman Vershynin.
Isoperimetric inequality on \mathbb{R}^n
Among all subsets A\subset \mathbb{R}^n with a given volume, the Euclidean ball has the minimal area.
That is, for any \epsilon>0, Euclidean balls minimize the volume of the $\epsilon$-neighborhood of A.
Where the volume of the $\epsilon$-neighborhood of A is defined as
A_\epsilon(A)\coloneqq \{x\in \mathbb{R}^n: \exists y\in A, \|x-y\|_2\leq \epsilon\}=A+\epsilon B_2^n
Here the \|\cdot\|_2 is the Euclidean norm. (The theorem holds for both geodesic metric on sphere and Euclidean metric on \mathbb{R}^n.)
Isoperimetric inequality on the sphere
Let \sigma_n(A) denotes the normalized area of A on n dimensional sphere S^n. That is \sigma_n(A)\coloneqq\frac{\operatorname{Area}(A)}{\operatorname{Area}(S^n)}.
Let \epsilon>0. Then for any subset A\subset S^n, given the area \sigma_n(A), the spherical caps minimize the volume of the $\epsilon$-neighborhood of A.
The above two inequalities is not proved in the Book High-dimensional probability.
To continue prove the theorem, we use sub-Gaussian concentration (Chapter 3 of High-dimensional probability by Roman Vershynin) of sphere \sqrt{n}S^n.
This will leads to some constant C>0 such that the following lemma holds:
The "Blow-up" lemma
Let A be a subset of sphere \sqrt{n}S^n, and \sigma denotes the normalized area of A. Then if \sigma\geq \frac{1}{2}, then for every t\geq 0,
\sigma(A_t)\geq 1-2\exp(-ct^2)
where A_t=\{x\in S^n: \operatorname{dist}(x,A)\leq t\} and c is some positive constant.
Proof of the Levy's concentration theorem
Proof:
Without loss of generality, we can assume that \eta=1. Let M denotes the median of f(X).
So \operatorname{Pr}[|f(X)\leq M|]\geq \frac{1}{2}, and \operatorname{Pr}[|f(X)\geq M|]\geq \frac{1}{2}.
Consider the sub-level set A\coloneqq \{x\in \sqrt{n}S^n: |f(x)|\leq M\}.
Since \operatorname{Pr}[X\in A]\geq \frac{1}{2}, by the blow-up lemma, we have
\operatorname{Pr}[X\in A_t]\geq 1-2\exp(-ct^2)
And since
\operatorname{Pr}[X\in A_t]\leq \operatorname{Pr}[f(X)\leq M+t]
Combining the above two inequalities, we have
\operatorname{Pr}[f(X)\leq M+t]\geq 1-2\exp(-ct^2)
Levy's concentration theorem in Metric Structures for Riemannian and Non-Riemannian Spaces by M. Gromov
Levy's concentration theorem (Gromov's version)
The Levy's lemma can also be found in Metric Structures for Riemannian and Non-Riemannian Spaces by M. Gromov.
3\frac{1}{2}.19The Levy concentration theory.
Theorem 3\frac{1}{2}.19 Levy concentration theorem:
An arbitrary 1-Lipschitz function f:S^n\to \mathbb{R} concentrates near a single value a_0\in \mathbb{R} as strongly as the distance function does.
That is
\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
where
\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.