Cheeger bound

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 $X$ be a finite set and let $K(x,y)$ be the transition probability for a reversible Markov chain on $X$. Assume this chain has stationary distribution $\pi$.

Define

$Q(x,y)=\pi (x)K(x,y)$

and for $A,B\subset X$ define

$Q(A\times B)=\sum _{x\in A,y\in B}Q(x,y).$

Define the constant $\mathrm{\Phi }$ as

$\mathrm{\Phi }=\underset{S\subset X,\pi (S)\le \frac{1}{2}}{\min }\frac{Q(S\times S^c)}{\pi (S)}.$

The operator $K,$ acting on the space of functions from $|X|$ to $\mathbb{ℝ}$, defined by

$(K\varphi )(x)=\sum _{y}K(x,y)\varphi (y)$

has eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$. It is known that ${\lambda }_1=1$. The Cheeger bound is a bound on the second largest eigenvalue ${\lambda }_2$.

Theorem (Cheeger bound):

$1-2\mathrm{\Phi }\le {\lambda }_2\le 1-\frac{{\mathrm{\Phi }}^2}{2}.$

See also

• Stochastic matrix
• Cheeger constant
• Conductance

References

• Cheeger, Jeff (1971). "A Lower Bound for the Smallest Eigenvalue of the Laplacian". Problems in Analysis: A Symposium in Honor of Salomon Bochner (PMS-31). Princeton University Press. pp. 195–200. doi:10.1515/9781400869312-013. ISBN 978-1-4008-6931-2. 
• Diaconis, Persi; Stroock, Daniel (1991). "Geometric Bounds for Eigenvalues of Markov Chains". The Annals of Applied Probability (Institute of Mathematical Statistics) 1 (1): 36–61. doi:10.1214/aoap/1177005980. ISSN 1050-5164. http://www.jstor.org/stable/2959624. Retrieved 2024-04-14.