4.4 KiB
4.4 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Cheeger bound | 1/1 | https://en.wikipedia.org/wiki/Cheeger_bound | reference | science, encyclopedia | 2026-05-05T12:21:47.049744+00:00 | kb-cron |
In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs. Let
X
{\displaystyle X}
be a finite set and let
K
(
x
,
y
)
{\displaystyle K(x,y)}
be the transition probability for a reversible Markov chain on
X
{\displaystyle X}
. Assume this chain has stationary distribution
π
{\displaystyle \pi }
. Define
Q
(
x
,
y
)
=
π
(
x
)
K
(
x
,
y
)
{\displaystyle Q(x,y)=\pi (x)K(x,y)}
and for
A
,
B
⊂
X
{\displaystyle A,B\subset X}
define
Q
(
A
×
B
)
=
∑
x
∈
A
,
y
∈
B
Q
(
x
,
y
)
.
{\displaystyle Q(A\times B)=\sum _{x\in A,y\in B}Q(x,y).}
Define the constant
Φ
{\displaystyle \Phi }
as
Φ
=
min
S
⊂
X
,
π
(
S
)
≤
1
2
Q
(
S
×
S
c
)
π
(
S
)
.
{\displaystyle \Phi =\min _{S\subset X,\pi (S)\leq {\frac {1}{2}}}{\frac {Q(S\times S^{c})}{\pi (S)}}.}
The operator
K
,
{\displaystyle K,}
acting on the space of functions from
|
X
|
{\displaystyle |X|}
to
R
{\displaystyle \mathbb {R} }
, defined by
(
K
ϕ
)
(
x
)
=
∑
y
K
(
x
,
y
)
ϕ
(
y
)
{\displaystyle (K\phi )(x)=\sum _{y}K(x,y)\phi (y)}
has eigenvalues
λ
1
≥
λ
2
≥
⋯
≥
λ
n
{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}}
. It is known that
λ
1
=
1
{\displaystyle \lambda _{1}=1}
. The Cheeger bound is a bound on the second largest eigenvalue
λ
2
{\displaystyle \lambda _{2}}
. Theorem (Cheeger bound):
1
−
2
Φ
≤
λ
2
≤
1
−
Φ
2
2
.
{\displaystyle 1-2\Phi \leq \lambda _{2}\leq 1-{\frac {\Phi ^{2}}{2}}.}
== See also == Stochastic matrix Cheeger constant Conductance
== References ==