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