发表机构
Stony Brook University（石溪大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出统一几何框架，整合多种可微嵌入诊断方法，证明积分视角不可约，实验验证其能准确估计单细胞嵌入可信性并区分不同类型嵌入。
AI 中文摘要
分析人员如何判断非线性降维嵌入是否可信？现有诊断方法仅能给出部分答案：投影 glyphs（ glyphs 可译为“ glyph 图”）表征局部敏感性，映射连续性分数衡量局部条件数，基于传输的分析揭示路径依赖的不一致性。然而这些方法看似互不关联，且未提供统一框架以理解它们何时一致。我们证明，这些方法均源自每个可微嵌入所诱导的单一几何对象，该嵌入既可以是通过优化隐式定义的，也可以是由学习到的映射显式定义的。该框架为嵌入提供两种互补的几何视角：微分视角解释局部行为，其一阶项可还原投影 glyphs，其二阶曲率量化其线性近似保持可靠的程度；积分视角沿高维路径追踪相同几何，确定嵌入是否仅依赖当前状态，还是也依赖到达该状态所走的路径。我们进一步证明，映射连续性是其他分析的前提条件。该框架对于源自嵌入几何的诊断具有理论完备性，且我们证明积分视角具有不可约性：无论在多少个点上进行任意阶导数的局部测量，都无法复现其检测到的内容。经典的基于秩的指标构成了基于有限尺度邻域关系的互补类别。在合成数据集和真实数据集上的实验验证了理论预测，证明了基于曲率的单细胞嵌入可信性估计的准确性，且积分分析能够以现有逐点诊断无法做到的方式，将单值嵌入与基于路径依赖优化的嵌入区分开来。
英文摘要
How can an analyst decide whether a nonlinear dimensionality reduction embedding can be trusted? Existing diagnostics provide only partial answers: projection glyphs characterize local sensitivity, map-continuity scores measure local conditioning, and transport-based analyses reveal path-dependent inconsistencies. However, these methods appear unrelated and provide no common framework for understanding when they agree or not. We show that they are all derived from a single geometric object induced by every differentiable embedding, whether defined implicitly through optimization or explicitly by a learned mapping. This framework provides two complementary geometric views of an embedding. The differential view explains local behavior: its first-order term recovers projection glyphs, while its second-order curvature quantifies how far their linear approximation remains reliable. The integral view follows the same geometry along high dimensional paths and determines whether an embedding depends only on the current state or also on the path taken to reach it. We further show that map-continuity is a prerequisite for the other analyses. The framework is theoretically complete for diagnostics derived from the embedding geometry, and we prove the integral view irreducible: no amount of local measurement at any number of points, to any order of derivative, reproduces what it detects. Classical rank-based metrics form a complementary class based on finite-scale neighborhood relationships. Experiments on synthetic and real datasets validate theoretical predictions, demonstrate accurate curvature-based trust estimates on single-cell embeddings, and show that the integral analysis distinguishes single-valued embeddings from path-dependent optimization-based embeddings in ways that existing pointwise diagnostics cannot.
Comments19 pages in total, 12 for the main and 7 for the appendices. 9 figures in the main and 7 in the appendices
