发表机构
Yale University; IE University（耶鲁大学; IE大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对流形假设下聚类数不确定性问题，提出基于流形的聚类方法，通过几何权衡形式化不确定性区间，量化而非强制解决聚类数模糊性。
AI 中文摘要
流形假设提出了一种自然的聚类准则：根据每个数据点所来源的流形分量对数据进行划分。两个分量是否可分离取决于一个几何权衡：分量之间的环境分离度与采样中的最大间隙之间的权衡。在实践中，这种权衡很少被明确评估，导致标准方法即使在数据不支持唯一答案时，也过度承诺于单一的聚类分配。我们通过将内在流形几何（体积增长和到达距离）与样本级量（填充距离和密度）相结合，形式化了这一权衡，从而为互k近邻图产生了一个阈值现象：当偏移与填充比率超过一个保守的上阈值时，分量分离得以保留；低于下阈值时，分量融合。这些阈值之间的间隙定义了一个几何不确定性区域，在该区域中，聚类的数量无法从数据中识别。然而，传统方法仍然寻求一个聚类数：扫描参数（工程方法）或拟合生成式混合模型（基于模型的方法）。我们不强制估计单一的聚类数量，而是提出基于流形的聚类（MBC），该方法返回一个明确的区间括注，以量化底层数据的不确定性。该括注作为一个经验校准的诊断工具：当支持单一分辨率时，它变窄；当多种分辨率共存时，它变宽；当检测不到分离结构时，它收缩为一。实验上，我们发现许多真实数据集位于不确定性区域内，而不是提供一个明确的答案。我们的结果表明，聚类数量的模糊性通常是内在的，应该被量化而不是被解决。
英文摘要
The manifold hypothesis suggests a natural criterion for clustering: partition data according to the manifold component from which each point is drawn. Whether two components are separable depends on a geometric tradeoff: the ambient separation between components versus the largest gap in sampling. In practice, this tradeoff is rarely assessed explicitly, leading standard methods to over-commit to a single clustering assignment even when the data do not support a unique answer. We formalize this tradeoff by combining intrinsic manifold geometry (volume growth and reach) with sample-level quantities (fill distance and density), yielding a threshold phenomenon for mutual-$k$-nearest-neighbor graphs: when the offset-to-fill ratio exceeds a conservative upper threshold, component separation is preserved; below a lower threshold, components fuse. The gap between these thresholds defines a geometric uncertainty zone in which the number of clusters is not identifiable from the data. Nevertheless, conventional approaches still seek one: sweeping parameters (an engineering approach) or fitting a generative mixture model (a model-based approach). Rather than forcing a single estimate of the number of clusters, we propose Manifold-Based Clustering (MBC), which returns an explicit bracket interval to quantify the underlying data uncertainty. This bracket acts as an empirically calibrated diagnostic: it narrows when a single resolution is supported, widens when multiple resolutions coexist, and collapses to one when no separated structure is detectable. Empirically, we find that many real datasets lie within the uncertainty zone rather than admitting one clear answer. Our results suggest that ambiguity in cluster number is often intrinsic, and should be quantified rather than resolved.
