21 KiB
Math 401, Fall 2025: Thesis notes, R2, Levy's concentration theorem and Levy's family
Progress: 2/5=40% (denominator and enumerator may change)
Levy's concentration theorem
Tip
This version of Levy's concentration theorem can be found in Geometry of Quantum states 15.84 and 15.85.
Our goal is to prove the generalized version of Levy's concentration theorem used in Hayden's work for $\eta$-Lipschitz functions.
Let f:S^{n-1}\to \mathbb{R} be a $\eta$-Lipschitz function. Let M_f denote the median of f and \langle f\rangle denote the mean of f. (Note this can be generalized to many other manifolds.)
Select a random point x\in S^{n-1} with n>2 according to the uniform measure (Haar measure). Then the probability of observing a value of f much different from the reference value is exponentially small.
\operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2})
\operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2})
Levy's concentration theorem via sub-Gaussian concentration
Tip
This version of Levy's concentration theorem can be found in High-dimensional probability 5.1.4.
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. But you can find it in the Appendix C of Gromov's book Metric Structures for Riemannian and Non-Riemannian Spaces.
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 via Levy family
Tip
This version of Levy's concentration theorem can be found in:
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}
a_0 is the Levy mean of function f, that is the level set of f^{-1}:\mathbb{R}\to S^n divides the sphere into equal halves, characterized by the following equality:
\mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
Detailed proof by Takashi Shioya.
The central idea is to draw the connection between the given three topological spaces,
S^{2n+1},CP^nand\mathbb{R}.
First, we need to introduce the following distribution and lemmas/theorems:
OBSERVATION
consider the orthogonal projection from \mathbb{R}^{n+1}, the space where S^n is embedded, to \mathbb{R}^k, we denote the restriction of the projection as \pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k. Note that \pi_{n,k} is a 1-Lipschitz function (projection will never increase the distance between two points).
We denote the normalized Riemannian volume measure on S^n(\sqrt{n}) as \sigma^n(\cdot), and \sigma^n(S^n(\sqrt{n}))=1.
Definition of Gaussian measure on \mathbb{R}^k
We denote the Gaussian measure on \mathbb{R}^k as \gamma^k.
d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx
x\in \mathbb{R}^k, \|x\|^2=\sum_{i=1}^k x_i^2 is the Euclidean norm, and dx is the Lebesgue measure on \mathbb{R}^k.
Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on \mathbb{R}^k with standard deviation 1.
Maxwell-Boltzmann distribution law
It is such a wonderful fact for me, that the projection of
n+1dimensional sphere with radius\sqrt{n}to\mathbb{R}^kis a Gaussian distribution asn\to \infty.
For any natural number k,
\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}\to \frac{d\gamma^k(x)}{dx}
where (\pi_{n,k})_*\sigma^n is the push-forward measure of \sigma^n by \pi_{n,k}.
In other words,
(\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
Proof
We denote the n dimensional volume measure on \mathbb{R}^k as \operatorname{vol}_k.
Observe that \pi_{n,k}^{-1}(x),x\in \mathbb{R}^k is isometric to S^{n-k}(\sqrt{n-\|x\|^2}), that is, for any x\in \mathbb{R}^k, \pi_{n,k}^{-1}(x) is a sphere with radius \sqrt{n-\|x\|^2} (by the definition of \pi_{n,k}).
So,
\begin{aligned}
\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}&=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(x))}{\operatorname{vol}_k(S^n(\sqrt{n}))}\\
&=\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}\\
\end{aligned}
as n\to \infty.
note that \lim_{n\to \infty}{(1-\frac{a}{n})^n}=e^{-a} for any a>0.
(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})
So
\begin{aligned}
\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
&=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
&=\frac{d\gamma^k(x)}{dx}
\end{aligned}
QED
Proof of the Levy's concentration theorem via the Maxwell-Boltzmann distribution law
We use the Maxwell-Boltzmann distribution law and Levy's isoperimetric inequality to prove the Levy's concentration theorem.
The goal is the same as the Gromov's version, first we bound the probability of the sub-level set of f by the \kappa_n(\epsilon) function by Levy's isoperimetric inequality. Then we claim that the \kappa_n(\epsilon) function is bounded by the Gaussian distribution.
Note, this section is not rigorous enough in sense of mathematics and the author should add sections about Levy family and observable diameter to make the proof more rigorous and understandable.
Proof
Let f:S^n\to \mathbb{R} be a 1-Lipschitz function.
Consider the two sets of points on the sphere S^n with radius \sqrt{n}:
\Omega_+=\{x\in S^n: f(x)\leq a_0-\epsilon\}, \Omega_-=\{x\in S^n: f(x)\geq a_0+\epsilon\}
Note that \Omega_+\cup \Omega_- is the whole sphere S^n(\sqrt{n}).
By the Levy's isoperimetric inequality, we have
\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))\leq \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_+))+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_-))
We define \kappa_n(\epsilon) as the following:
\kappa_n(\epsilon)=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))}{\operatorname{vol}_k(S^n(\sqrt{n}))}=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
By the Levy's isoperimetric inequality, and the Maxwell-Boltzmann distribution law, we have
\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
Levy's Isoperimetric inequality
This section is from the Appendix
C_+of Gromov's book Metric Structures for Riemannian and Non-Riemannian Spaces.
Not very edible for undergraduates.
Differential Geometry
This section is designed for stupids like me skipping too much essential materials in the book.
This part might be extended to a separate note, let's check how far we can go from this part.
References:
[Introduction to Smooth Manifolds by John M. Lee]
Manifold
Unexpectedly, a good definition of the manifold is defined in the topology I.
Check section 36. This topic extends to a wonderful chapter 8 in the book where you can hardly understand chapter 2.
Definition of m-manifold
An $m$-manifold is a Hausdorff space X with a countable basis (second countable) such that each point of x of X has a neighborhood homeomorphic to an open subset of \mathbb{R}^m.
Example of second countable space
Let X=\mathbb{R} and \mathcal{B}=\{(a,b)|a,b\in \mathbb{R},a<b\} (collection of all open intervals with rational endpoints).
Since the rational numbers are countable, so \mathcal{B} is countable.
So \mathbb{R} is second countable.
Likewise, \mathbb{R}^n is also second countable.
Example of manifold
1-manifold is a curve and 2-manifold is a surface.
Theorem of imbedded space
If X is a compact $m$-manifold, then X can be imbedded in \mathbb{R}^n for some n.
This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers.
Smooth manifolds and Lie groups
This section is waiting for the completion of book Introduction to Smooth Manifolds by John M. Lee.
Partial derivatives
Let U\subseteq \mathbb{R}^n and f:U\to \mathbb{R}^n be a map.
For any a=(a_1,\cdots,a_n)\in U, j\in \{1,\cdots,n\}, the $j$-th partial derivative of F at a is defined as
\begin{aligned}
\frac{\partial f}{\partial x_j}(a)&=\lim_{h\to 0}\frac{f(a_1,\cdots,a_j+h,\cdots,a_n)-f(a_1,\cdots,a_j,\cdots,a_n)}{h} \\
&=\lim_{h\to 0}\frac{f(a+he_j)-f(a)}{h}
\end{aligned}
Continuously differentiable maps
Let U\subseteq \mathbb{R}^n and f:U\to \mathbb{R}^n be a map.
If for any j\in \{1,\cdots,n\}, the $j$-th partial derivative of f is continuous at a, then f is continuously differentiable at a.
If \forall a\in U, \frac{\partial f}{\partial x_j} exists and is continuous at a, then f is continuously differentiable on U. or C^1 map. (Note that C^0 map is just a continuous map.)
Smooth maps
A function f:U\to \mathbb{R}^n is smooth if it is of class C^k for every k\geq 0 on U. Such function is called a diffeomorphism if it is also a bijection and its inverse is also smooth.
Charts
Let M be a smooth manifold. A chart is a pair (U,\varphi) where U\subseteq M is an open subset and \varphi:U\to \hat{U}\subseteq \mathbb{R}^n is a homeomorphism (a continuous bijection map and its inverse is also continuous).
If p\in U and \varphi(p)=0, then we say that p is the origin of the chart (U,\varphi).
For p\in U, we note that the continuous function \varphi(p)=(x_1(p),\cdots,x_n(p)) gives a vector in \mathbb{R}^n. The (x_1(p),\cdots,x_n(p)) is called the local coordinates of p in the chart (U,\varphi).
Atlas
Let M be a smooth manifold. An atlas is a collection of charts \mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I} such that M=\bigcup_{\alpha\in I} U_\alpha.
An atlas is said to be smooth if the transition maps \phi_\alpha\circ \phi_\beta^{-1}:\phi_\beta(U_\alpha\cap U_\beta)\to \phi_\alpha(U_\alpha\cap U_\beta) are smooth for all \alpha, \beta\in I.
Smooth manifold
A smooth manifold is a pair (M,\mathcal{A}) where M is a topological manifold and \mathcal{A} is a smooth atlas.
Fundamental group
A fundamental group of a point p in a topological space X is the group of all paths (continuous map f:I\to X, I=[0,1]\subseteq \mathbb{R}) from p to p.
- Product defined as composition of paths.
- Identity element is the constant path from
ptop. - Inverse is the reverse path.
smooth local coordinate representations
If M is a smooth manifold, then any chart (U,\varphi) contained in the given maximal smooth atlas is called a smooth chart, and the map \varphi is called a smooth coordinate map because it gives a coordinate
Lie group
Lie group is a group (satisfying group axioms: closure, associativity, identity, inverses) that is also a smooth manifold. with the operator m:G\times G\to G, and the inverse operation i:G\to G that are both smooth.
In short, a Lie group is a group that is also a smooth manifold with map G\times G\to G given by (g,h)\mapsto gh^-1 that is smooth.
Example of Lie group
The general linear group GL(n,\mathbb{R}) is the group of all n\times n invertible matrices over \mathbb{R}.
This is a Lie group since
- Multiplication is a smooth map
GL(n,\mathbb{R})\times GL(n,\mathbb{R})\to GL(n,\mathbb{R})since it is a polynomial map. - Inverse is a smooth map
GL(n,\mathbb{R})\to GL(n,\mathbb{R})by cramer's rule.
If G is a Lie group, then any open subgroup (with subgroup topology and open set in G) H of G is also a Lie group.
Translation map on Lie group
If G is a Lie group, then the translation map L_g:G\to G given by L_g(h)=gh and R_g:G\to G given by R_g(h)=hg are both smooth and are diffeomorphisms on G.
Derivation and tangent vectors
The directional derivative of a geometric tangent vector v_a\in \mathbb{R}^n_a yields a map D_v\vert_a:C^\infty(\mathbb{R}^n)\to \mathbb{R} given by the formula
D_v\vert_a(f)=D_v f(a)=\frac{d}{dt}\bigg\vert_{t=0}f(a+tv_a)
Note that this is a linear over \mathbb{R}, and satisfies the product rule.
D_v\vert_a(f\cdot g)=f(a)D_v\vert_a(g)+g(a)D_v\vert_a(f)
We can generalize this representation to the following definition:
If a is a point of \mathbb{R}^n, then a derivation at $a$ is a linear map w:C^\infty(\mathbb{R}^n)\to \mathbb{R} such that it is linear over \mathbb{R} and satisfies the product rule.
w(f\cdot g)=w(f)\cdot g(a)+f(a)\cdot w(g)
Let T_a\mathbb{R}^n denote the set of all derivations of C^\infty(\mathbb{R}^n) at a. So T_a\mathbb{R}^n is a vector space over \mathbb{R}.
(w_1+w_2)(f)=w_1(f)+w_2(f),\quad (cw)(f)=c(w(f))
Some key properties are given below and check the proof in the book for details.
- If
fis a constant function, thenw(f)=0. - If
f(a)=g(a)=0, thenw(f\cdot g)=0. - For each geometric tangent vector
v_a\in \mathbb{R}^n_a, the mapD_v\vert_a:C^\infty(\mathbb{R}^n)\to \mathbb{R}is a derivation ata. - The map
v_a\mapsto D_v\vert_ais an isomorphism of vector spaces from\mathbb{R}^n_atoT_a\mathbb{R}^n.
Tangent vector on Manifolds
Let M be a smooth manifold. Let p\in M. A tangent vector to M at $p$ is a derivation at p if it satisfies:
v(f\cdot g)=f(p)vg+g(p)vf\prod \text{ for all } f,g\in C^\infty(M)
The set of all derivations of C^\infty(M) at p is denoted by T_pM is called tangent space to M at p. An element of T_pM is called a tangent vector to M at p.
Tangent bundle
We define the tangent bundle of M as the disjoint union of all the tangent spaces:
TM=\bigsqcup_{p\in M} T_pM
We write the element in TM as pair (p,v) where p\in M and v\in T_pM.
The tangent bundle comes with a natural projection map \pi:TM\to M given by \pi(p,v)=p.
Section of map
If \pi:M\to N is any continuous map, a section of $\pi$ is a continuous right inverse of \pi. For example \sigma:N\to M is a section of \pi if \sigma\circ \pi=Id_N.
Vector field
A vector field on M is a section of the map \pi:TM\to M.
More concretely, a vector field is a continuous map X:M\to TM, usually written p\mapsto X_p, with property that
\pi\circ X=Id_M
That is a map from element on the manifold to the tangent space of the manifold.
Riemannian manifolds and geometry
Riemannian metric
A Riemannian metric is a smooth assignment of an inner product to each tangent space T_pM of the manifold.
More formally, let M be a smooth manifold. A Riemannian metric on M is a smooth covariant 2-tensor field g\in \mathcal{T}^2(M) whose value g_p at each p\in M is an inner product on T_p M.
Thus g is a symmetric 2-tensor field that is positive definite in the sense that g_p(v,v)\geq 0 for each p\in M and each v\in T_p M, with equality if and only if v=0.
Riemannian metric exists in great abundance.
A good news for smooth manifold is that every smooth manifold admits a Riemannian metric.
Example of Riemannian metrics
An example of Riemannian metric is the Euclidean metric, the bilinear form of d(p,q)=\|p-q\|_2 on \mathbb{R}^n.
More formally, the Riemannian metric \overline{g} on \mathbb{R}^n at each x\in \mathbb{R}^n , for v,w\in T_x \mathbb{R}^n with stardard coordinates (x^1,\ldots,x^n) as v=\sum_{i=1}^n v_i \partial_x^i and $w=\sum_{
Riemannian manifolds
A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which is a smooth assignment of an inner product to each tangent space T_pM of the manifold.
More formally, a Riemannian manifold is a pair (M,g), where M is a smooth manifold and g is a specific choice of Riemannian metric on M.
An example of Riemannian manifold is the sphere \mathbb{C}P^n.
Notion of Connection
A connection is a way to define the directional derivative of a vector field along a curve on a Riemannian manifold.
For every p\in M, where M denote the manifold, suppose M=\mathbb{R}^n, then let X=(f_1,\cdots,f_n) be a vector field on M. The directional derivative of X along the point p is defined as
D_VX=\lim_{h\to 0}\frac{X(p+h)-X(p)}{h}
Notion of Curvatures
Note
Geometrically, the curvature of the manifold is radius of the tangent sphere of the manifold.
Nabla notation and Levi-Civita connection
Fundamental theorem of Riemannian geometry
Let (M,g) be a Riemannian or pseudo-Riemannian manifold (with or without boundary). There exist sa unique connection \nabla on TM that is compatible with g and symmetric. It is called the Levi-Civita connection of g (or also, when g is a positive definite, the Riemannian connection).