# Computationally Efficient Estimation of the Spectral Gap of a Markov Chain

@article{Combes2019ComputationallyEE, title={Computationally Efficient Estimation of the Spectral Gap of a Markov Chain}, author={Richard Combes and Mikael Touati}, journal={Abstracts of the 2019 SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer Systems}, year={2019} }

We consider the problem of estimating from sample paths the absolute spectral gap 1-λ⋆ of a reversible, irreducible and aperiodic Markov chain (Xt)t∈N over a finite state space Ω. We propose the UCPI (Upper Confidence Power Iteration) algorithm for this problem, a low-complexity algorithm which estimates the spectral gap in time O(n) and memory space O((ln n)2 given n samples. This is in stark contrast with most known methods which require at least memory space O(|Ω|), so that they cannot be…

## 3 Citations

Mixing Time Estimation in Ergodic Markov Chains from a Single Trajectory with Contraction Methods

- MathematicsALT
- 2020

This new approach based on contraction methods is the first that aims at directly estimating the mixing time of an ergodic Markov chain up to multiplicative small universal constants instead of t, by introducing a generalized version of Dobrushin's contraction coefficient $\kappa_{\mathsf{gen}}$, which is shown to control themix time regardless of reversibility.

Fiedler Vector Approximation via Interacting Random Walks

- Mathematics, Computer ScienceAbstracts of the 2020 SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer Systems
- 2020

This paper develops a framework in which a stochastic process is constructed based on a set of interacting random walks on a graph and shows that a suitably scaled version of the stochastics process converges to the Fiedler vector for a sufficiently large number of walks.

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This paper develops a framework in which a stochastic process based on a set of interacting random walks on a graph is constructed and it is shown that a suitably scaled version of the stochastics process converges to the Fiedler vector for a sufficiently large number of walks.

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