🤖 AI Summary
Non-asymptotic mixing-time analysis of finite-state ergodic Markov chains—both reversible and irreversible—remains challenging, particularly due to the lack of a unified convergence characterization for irreversible chains.
Method: We develop an operator-theoretic framework based on orthogonal projections of the transition operator in the ℓ²(π) space, using matrix norms to quantify convergence rates.
Contribution/Results: We establish, for the first time, submultiplicativity of pointwise χ²-divergence for irreversible chains, yielding explicit, computable bounds dependent on spectral structure, algebraic–geometric multiplicity gaps, and condition numbers of similarity transformations. We apply this framework to momentum-based samplers, revealing their diffusive behavior via hypercontractivity and regression analysis. For irreversible triangular random walks on graphs, we derive the tightest known mixing-time bound to date and demonstrate near-optimal O(n¹·⁹⁷) convergence even under V-shaped target distributions.
📝 Abstract
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 $ell^{2}(pi)$, where $pi$ 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 $chi^{2}(k)$ 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, $nlog{n}$ iterations suffice for $chi^{2}$ 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 $ au_{rel}=|sum_{k=0}^{infty}P^{k}U_{perp 1}|_{ell^{2}(pi)}$.