Urysohn 阶梯：用于可扩展持续学习的递归度量压缩
The Urysohn Ladder: Recursive Metric Contraction for Scalable Continual Learning
AI总结：
提出 Urysohn 阶梯，通过递归度量压缩将无限环境搜索转化为有界内在导航，实现可分离性、有界容量、稳定性和可扩展性，解决持续学习中的灾难性干扰。
摘要：
持续学习系统面临一个基本的几何障碍：随着经验在固定容量流形上积累，覆盖数随时间线性增长，最终迫使表示重叠和灾难性干扰。现有方法通过扩展——通过核、过参数化或重放投影到更高维空间——来解决这个问题。我们认为解决方案恰恰相反：压缩。我们将抽象形式化为 Urysohn 阶梯，这是一个商映射的层次结构，递归地将已验证的度量邻域折叠为紧凑的令牌，将无界的环境空间搜索转化为低维内在支架上的有界导航。几何上，每个折叠的令牌充当一个捷径——一个极端度量压缩的区域，连接遥远的经验，就像表示流形中的虫洞。我们建立了四个结果，共同保证了可分离性（度量压缩使非线性纠缠结构在每个商层次上线性可分，并且这种可分性忠实地传播到整个层次结构）、有界容量（每个商层次的覆盖数保持 O(1)，与流长度无关）、稳定性（奇偶划分的流/支架子空间实现了无界可塑性而无灾难性干扰）和可扩展性（推理成本与商距离成比例，而非环境距离）。我们通过预训练模型和真实世界数据集实证验证了每个主张。此外，我们通过支架摊销展示了 Urysohn 阶梯在可扩展持续学习中的潜力。
英文摘要：
Continual learning systems face a fundamental geometric obstacle: as experience accumulates on a fixed-capacity manifold, covering numbers grow linearly with time, eventually forcing representational overlap and catastrophic interference. Prevailing approaches attack this problem by \emph{expansion} - projecting into higher-dimensional spaces via kernels, overparameterization, or replay. We argue the solution is the opposite: \emph{contraction}. We formalize abstraction as the \textbf{Urysohn Ladder}, a hierarchy of quotient maps that recursively collapse validated metric neighborhoods into compact tokens, converting unbounded ambient-space search into bounded navigation on a low-dimensional intrinsic scaffold. Geometrically, each collapsed token acts as a shortcut - a region of extreme metric contraction that bridges distant experiences, much like a wormhole in the representational manifold. We establish four results that collectively guarantee \emph{separability} (metric contraction renders nonlinearly entangled structure linearly separable at each quotient level, and this separability propagates faithfully through the entire hierarchy), \emph{bounded capacity} (covering numbers remain $O(1)$ per quotient level, independent of stream length), \emph{stability} (parity-partitioned flow/scaffold subspaces enable unbounded plasticity without catastrophic interference), and \emph{scalability} (inference cost scales with quotient distance, not ambient distance). We validate each claim empirically with pretrained models and real-world datasets. Moreover, we demonstrate the potential of Urysohn Ladder for scalable continual learning via scaffold amortization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
