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<p>We study how the choice of statistical geometry used to quantify “distance to stationarity” shapes the observed phase structure of non-Markovian relaxation in finite-state learning and inference dynamics. Starting from lattice Metropolis dynamics with a prescribed stationary distribution, we consider two standard Markovian embeddings that realize exponential-memory and Erlang-2 memory kernels. We compare two inequivalent geometries on the simplex: the information-theoretic Kullback–Leibler (KL) divergence and a geometry-aware entropic optimal-transport (OT) cost induced by a ground metric on the discrete state space. To enable a fair comparison across geometries, we define convergence times for both KL and OT via the same scale-free residual criterion and construct phase diagrams over noise and memory parameters. While overshoot regimes are qualitatively robust, the apparent convergence landscape can shift substantially when measured by entropic OT. A difference-map analysis of OT- and KL-based convergence times identifies metric-dependent transient regimes where phase boundaries inferred from divergence geometry do not coincide with those inferred from transport geometry. These results clarify which non-Markovian transient effects are kernel-induced and robust, and which reflect the underlying choice of statistical geometry.</p>

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Metric-dependent transients in non-Markovian relaxation: Kullback–Leibler vs entropic optimal transport

  • 期刊论文

  • 发表日期：

    2026年3月21日

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  • Koichi Nakagawa

摘要

We study how the choice of statistical geometry used to quantify “distance to stationarity” shapes the observed phase structure of non-Markovian relaxation in finite-state learning and inference dynamics. Starting from lattice Metropolis dynamics with a prescribed stationary distribution, we consider two standard Markovian embeddings that realize exponential-memory and Erlang-2 memory kernels. We compare two inequivalent geometries on the simplex: the information-theoretic Kullback–Leibler (KL) divergence and a geometry-aware entropic optimal-transport (OT) cost induced by a ground metric on the discrete state space. To enable a fair comparison across geometries, we define convergence times for both KL and OT via the same scale-free residual criterion and construct phase diagrams over noise and memory parameters. While overshoot regimes are qualitatively robust, the apparent convergence landscape can shift substantially when measured by entropic OT. A difference-map analysis of OT- and KL-based convergence times identifies metric-dependent transient regimes where phase boundaries inferred from divergence geometry do not coincide with those inferred from transport geometry. These results clarify which non-Markovian transient effects are kernel-induced and robust, and which reflect the underlying choice of statistical geometry.