Computationally Efficient Estimation of the Spectral Gap of a Markov Chain
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We consider the problem of estimating from sample paths the absolute spectral gap γ_* of a reversible, irreducible and aperiodic Markov chain (X_t)_t ∈N over a finite state 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(( 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 applied to large state spaces. Furthermore, UCPI is amenable to parallel implementation.
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