Non Asymptotic Mixing Time Analysis of Non-Reversible Markov Chains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908626715672576 |
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| author | Naeem, Muhammad Abdullah |
| author_facet | Naeem, Muhammad Abdullah |
| contents | We introduce a unified operator-theoretic framework for analyzing mixing times of finite-state ergodic Markov chains that applies to both reversible and non-reversible dynamics. The central object in our analysis is the projected transition operator $PU_{\perp 1}$, where $P$ is the transition kernel and $U_{\perp 1}$ is orthogonal projection onto mean-zero subspace in $\ell^{2}(π)$, where $π$ is the stationary distribution. We show that explicitly computable matrix norms of $(PU_{\perp 1})^k$ gives non-asymptotic mixing times/distance to stationarity, and bound autocorrelations at lag $k$. We establish, for the first time, submultiplicativity of pointwise chi-squared divergence in the general non-reversible case. We provide for all times $χ^{2}(k)$ bounds based on the spectrum of $PU_{\perp 1}$, i.e., magnitude of its distinct non-zero eigenvalues, discrepancy between their algebraic and geometric multiplicities, condition number of a similarity transform, and constant coming from smallest atom of stationary distribution(all scientifically computable). Furthermore, for diagonalizable $PU_{\perp 1}$, we provide explict constants satisfying hypocoercivity phenomenon for discrete time Markov Chains. Our framework enables direct computation of convergence bounds for challenging non-reversible chains, including momentum-based samplers for V-shaped distributions. We provide the sharpest known bounds for non-reversible walk on triangle. Our results combined with simple regression reveals a fundamental insight into momentum samplers: although for uniform distributions, $n\log{n}$ iterations suffice for $χ^{2}$ mixing, for V-shaped distributions they remain diffusive as $n^{1.969}\log{n^{1.956}}$ iterations are sufficient. The framework shows that for ergodic chains relaxation times $τ_{rel}=\|\sum_{k=0}^{\infty}P^{k}U_{\perp 1}\|_{\ell^{2}(π)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non Asymptotic Mixing Time Analysis of Non-Reversible Markov Chains Naeem, Muhammad Abdullah Probability Computation We introduce a unified operator-theoretic framework for analyzing mixing times of finite-state ergodic Markov chains that applies to both reversible and non-reversible dynamics. The central object in our analysis is the projected transition operator $PU_{\perp 1}$, where $P$ is the transition kernel and $U_{\perp 1}$ is orthogonal projection onto mean-zero subspace in $\ell^{2}(π)$, where $π$ is the stationary distribution. We show that explicitly computable matrix norms of $(PU_{\perp 1})^k$ gives non-asymptotic mixing times/distance to stationarity, and bound autocorrelations at lag $k$. We establish, for the first time, submultiplicativity of pointwise chi-squared divergence in the general non-reversible case. We provide for all times $χ^{2}(k)$ bounds based on the spectrum of $PU_{\perp 1}$, i.e., magnitude of its distinct non-zero eigenvalues, discrepancy between their algebraic and geometric multiplicities, condition number of a similarity transform, and constant coming from smallest atom of stationary distribution(all scientifically computable). Furthermore, for diagonalizable $PU_{\perp 1}$, we provide explict constants satisfying hypocoercivity phenomenon for discrete time Markov Chains. Our framework enables direct computation of convergence bounds for challenging non-reversible chains, including momentum-based samplers for V-shaped distributions. We provide the sharpest known bounds for non-reversible walk on triangle. Our results combined with simple regression reveals a fundamental insight into momentum samplers: although for uniform distributions, $n\log{n}$ iterations suffice for $χ^{2}$ mixing, for V-shaped distributions they remain diffusive as $n^{1.969}\log{n^{1.956}}$ iterations are sufficient. The framework shows that for ergodic chains relaxation times $τ_{rel}=\|\sum_{k=0}^{\infty}P^{k}U_{\perp 1}\|_{\ell^{2}(π)}$. |
| title | Non Asymptotic Mixing Time Analysis of Non-Reversible Markov Chains |
| topic | Probability Computation |
| url | https://arxiv.org/abs/2511.02265 |