二维和三维中扩散目标的贝叶斯跟踪
Bayesian Tracking of a Diffusing Target in Two and Three Dimensions
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中文总结 AI 辅助
该研究分析二维和三维中扩散目标的贝叶斯跟踪的相结构，发现二维贝叶斯最优跟踪总能成功、次优推理或致失败，三维存在成功及两种失败模式，相关结果适用于其他表面生长或定向聚合物相变。
中文摘要 AI 辅助
我们研究由带噪声的分布式传感器阵列监测的扩散目标的贝叶斯跟踪问题。基于早期与带移动缺陷的KPZ生长（或等价的定向聚合物钉扎问题）的映射，我们确定了超越此前研究的一维情形的相结构，涵盖贝叶斯最优推理和次优推理两种情况。在二维（d=2）中，理论分析和数值模拟均给出成功跟踪相与失败相之间的脱钉转变；弱耦合重整化群（RG）表明，贝叶斯最优跟踪在二维中总是成功，但过度自信的（次优）推理可能导致失败。在三维（d=3）中，跟踪可成功，或以两种不同方式失败：后验概率分布可能离域化（无检测），或可能尖锐定域化但位置错误（假检测）。这两种情况对应对数后验的Edwards-Wilkinson或Kardar-Parisi-Zhang统计。在二维参数相图中，这三个相在贝叶斯最优线上的一个类西森（Nishimori）多临界点处交汇。（仅模型误设即可驱动脱钉进入任一非钉扎相：信心不足导致扩散型失败，过度自信导致定域化但错误的失败。）我们通过数值方法和重整化群论据分析各相之间的转变，表明其中一些转变具有常规ε展开无法描述的非常规临界行为，定向聚合物的近期严格结果暗示了一种替代场景。我们的许多结果，包括钉扎转变处的标度关系和RG流结果，与其他涉及2+1D或3+1D中表面生长或定向聚合物的相变相关。
英文摘要
We study Bayesian tracking of a diffusing target monitored by a noisy distributed sensor array. Building on an earlier mapping to KPZ growth with a moving defect (or an equivalent directed polymer pinning problem) we determine the phase structure, beyond the previously-studied one-dimensional case, for both Bayes-optimal and suboptimal inference. In $d=2$, theoretical analysis and numerical simulations both give a depinning transition between a successful tracking phase and a failure phase. Weak-coupling RG shows that Bayes-optimal tracking is always successful in $d=2$, but failure can arise from overconfident (suboptimal) inference. In $d=3$, tracking can succeed, or can fail in two distinct ways: the posterior probability distribution may delocalize (no detection), or may become sharply localized, but at the wrong position (a false detection). The two possibilities correspond to Edwards-Wilkinson or Kardar-Parisi-Zhang statistics for the log-posterior. The three phases meet at a Nishimori-like multicritical point on a Bayes-optimal line in a two-parameter phase diagram. (Model misspecification alone can drive depinning into either unpinned phase: underconfidence gives diffuse failure, while overconfidence gives localized-but-wrong failure.) We analyze the transitions between the various phases numerically and with renormalization group arguments. We show that some of these have unusual critical behavior, which the conventional $ε$ expansion fails to describe. Recent rigorous results for directed polymers indicate an alternative scenario. Many of our results, including a scaling relation for exponents at pinning transitions and results for RG flows, are relevant to other phase transitions that involve surface growth or directed polymers in 2+1D or 3+1D.
发表机构
- Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎高等师范学院与巴黎文理研究大学）
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