Singular-value gap of nonreversible Markov processes

Abstract: We consider a generalization of the spectral gap of reversible Markov generators to nonreversible processes, following the recent work arXiv:2310.10876 on nonreversible finite-state Markov chains. Extending Chatterjee's observations, we find that this spectral quantity that we call the \textit{singular-value gap} characterizes the convergence of empirical averages, providing upper and lower bounds for finite-time variance uniformly over $L^2$-functions. A key observation is that when the singular-value gap is positive, the generator is invertible on the $L^2$-orthogonal complement of constant functions. In particular, the Poisson equation $-Lf = g$ can be solved, which enables our proof and connects our results to asymptotic variance and associated central limit theorems. We also compare the singular-value gap with the spectral gap of the reversibilized process, the mixing time in total-variation distance, and the Cheeger constant. Several examples are provided throughout the text. Among other potential applications of the singular-value gap, these examples illustrate that a positive singular-value gap can help with variance reduction for observable classes in MCMC sampling, uncover slow-mixing mechanisms, and certify convergence of empirical averages for diffusion operators with complicated spectrum.