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}
}
  • Richard Combes, M. Touati
  • Published 20 June 2019
  • Mathematics, Computer Science
  • Abstracts of the 2019 SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer Systems
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… 
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References

SHOWING 1-10 OF 13 REFERENCES
Estimating the spectral gap of a trace-class Markov operator
The utility of a Markov chain Monte Carlo algorithm is, in large part, determined by the size of the spectral gap of the corresponding Markov operator. However, calculating (and even approximating)
Mixing Time Estimation in Reversible Markov Chains from a Single Sample Path
TLDR
This article provides the first procedure for computing a fully data-dependent interval that traps the mixing time of a finite reversible ergodic Markov chain at a prescribed confidence level, and does not require the knowledge of any parameters of the chain.
Spectrum Estimation from Samples
TLDR
This work proposes a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix and provides finite--sample bounds on the expected error of the recovered eigen Values, which imply that the estimator is asymptotically consistent as the dimensionality of the distribution and sample size tend towards infinity.
Coupling from the past: A user's guide
TLDR
The Markov chain Monte Carlo method is modified so as to remove all bias in the output resulting from the biased choice of an initial state for the chain, and this method is called Coupling From The Past (CFTP).
Matrix Norm Estimation from a Few Entries
TLDR
A novel unbiased estimator based on counting small structures in a graph and provide guarantees that match its empirical performances is introduced and shows that Schatten norms can be recovered accurately from strictly smaller number of samples compared to what is needed to recover the underlying low-rank matrix.
Random walks and an O * ( n 5 ) volume algorithm for convex bodies
TLDR
This algorithm introduces three new ideas: the use of the isotropic position (or at least an approximation of it) for rounding; the separation of global obstructions and local obstructions for fast mixing; and a stepwise interlacing of rounding and sampling.
Probability Inequalities for sums of Bounded Random Variables
Abstract Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S
Information, Physics, and Computation
TLDR
The approach focuses on large random instances, adopting a common probabilistic formulation in terms of graphical models, and presents message passing algorithms like belief propagation and survey propagation, and their use in decoding and constraint satisfaction solving.
Hit-and-run mixes fast
TLDR
It is shown that the “hit-and-run” algorithm for sampling from a convex body K mixes in time O*(n2R2/r2), where R and r are the radii of the inscribed and circumscribed balls of K and the bound is best possible in terms of R,r and n.
Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations
TLDR
This paper presents a meta-anatomy of the trace of a matrix function and its applications in machine learning, computational physics, and mathematics.
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