局域化景观作为异质系统的算子诱导几何
Localization Landscapes as Operator-Induced Geometry for Heterogeneous Systems
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中文总结 AI 辅助
本文提出局域化景观作为算子诱导几何，通过有效势揭示异质系统中的隐藏结构，解释低能模支撑并增强注意力局域性，其作为归纳偏置在约束场景有效但任务依赖。
中文摘要 AI 辅助
异质算子通常通过隐藏的势垒、势阱和弱连通隔室来组织响应，这些结构无法仅由欧几里得距离或原始图连通性忠实地描述。我们通过局域化景观（由单次源求解获得）及相应的有效势 $W_{\mathrm{eff}} = 1/u$ 来研究这一结构。从离散薛定谔型算子和异质反应-扩散算子出发，我们展示了 $W_{\mathrm{eff}}$ 如何诱导有效势阱、盆地划分和Agmon型邻域，这些作为算子感知的几何结构用于局域化和输运。该几何结构解释了低能本征模的空间支撑，在固定图上产生自适应感受野，锐化类注意力交互中的局域性，并产生对瓶颈敏感的扩散调制。我们还包含一个具有强隔室化系数的有限元反应-扩散示例，证明同一框架可扩展到简单网格算子之外的异质连续介质系统。结果支持一个精确结论：当相关性由约束、势垒和低能可达性控制时，局域化景观几何是一种强归纳偏置，但其益处依赖于任务而非普遍适用。
英文摘要
Heterogeneous operators often organize response through hidden barriers, wells, and weakly communicating compartments that are not faithfully described by Euclidean distance or raw graph connectivity alone. We study this structure through the localization landscape, obtained from a single source solve, and the associated effective potential $W_{\mathrm{eff}} = 1/u$. Starting from both discrete Schrödinger-type operators and heterogeneous reaction-diffusion operators, we show how $W_{\mathrm{eff}}$ induces effective wells, basin partitions, and Agmon-type neighborhoods that serve as an operator-aware geometry for localization and transport. This geometry explains the spatial support of low-energy eigenmodes, produces adaptive receptive fields on a fixed graph, sharpens locality in attention-like interactions, and yields bottleneck-sensitive modulation of diffusion. We also include a finite-element reaction-diffusion example with strongly compartmentalized coefficients, demonstrating that the same framework extends beyond simple grid operators to heterogeneous continuum systems. The results support a precise conclusion: localization-landscape geometry is a strong inductive bias when relevance is controlled by confinement, barriers, and low-energy accessibility, but its benefits are task-dependent rather than universal.
发表机构
- Columbia University（哥伦比亚大学）
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