Cheeger inequalities for transient Markov chains
نویسنده
چکیده
We construct upper and lower bounds for Cheeger-type constants for transient Markov chains. We first treat the situation where there is a detailed balance condition and obtain bounds that rely exclusively on the first and second eigenvalues of the substochastic transition matrix. Secondly, we consider general substochastic transition matrices and develop bounds in this broader setting. The efficacy of the bounds are illustrated via examples.
منابع مشابه
Eigenvalues of non-reversible Markov chains: their connection to mixing times, reversible Markov chains, and Cheeger inequalities
We show a lower bound on mixing time for a non-reversible Markov chain in terms of its eigenvalues. This is used to show a bound on the real part of the complex-valued eigenvalues in terms of the realvalued eigenvalues of a related reversible chain, and likewise to bound the second largest magnitude eigenvalue. A myriad of Cheeger-like inequalities also follow for non-reversible chains, which e...
متن کاملGeneralized Cheeger inequalities for eigenvalues of non-reversible Markov chains
We show lower bounds for the smallest non-trivial eigenvalue, and smallest real portion of an eigenvalue, of the Laplacian of a non-reversible Markov chain in terms of an Evolving set quantity. A myriad of Cheeger-like inequalities follow for non-reversible chains, which even in the reversible case sharpen previously known results. The same argument also produces a new Cheeger-like inequality f...
متن کاملApplied Probability Trust (20 October 2015) CHEEGER INEQUALITIES FOR ABSORBING MARKOV CHAINS
We construct Cheeger-type bounds for the second eigenvalue of a substochastic transition probability matrix in terms of the Markov chain’s conductance and metastability (and vice-versa) with respect to its quasi-stationary distribution, extending classical results for stochastic transition matrices.
متن کاملTransportation-information Inequalities for Markov Processes (ii) : Relations with Other Functional Inequalities
We continue our investigation on the transportation-information inequalities WpI for a symmetric markov process, introduced and studied in [14]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure μ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion pr...
متن کاملSharp edge, vertex, and mixed Cheeger type inequalities for finite Markov kernels
We show how the evolving set methodology of Morris and Peres can be used to show Cheeger inequalities for bounding the spectral gap of a finite Markov kernel. This leads to sharp versions of several previous Cheeger inequalities, including ones involving edge-expansion, vertex-expansion, and mixtures of both. A bound on the smallest eigenvalue also follows.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013