揭示数据维度与拓扑结构的关系，发现低维流形可被有效捕捉。
How high is `high'? Rethinking the roles of dimensionality in topological data analysis and manifold learning
- 提出广义Hanson-Wright不等式，解析三类维度的几何作用。
- 证明当p_int ≫ log n时，拓扑特征可准确揭示流形结构。
- 首次发现网格细胞活动与物理空间等距，具几何保真性。
我们提出了一个广义的Hanson-Wright不等式，并据此对点云数据的几何特性提供了新的统计洞察。在一般随机函数模型下，厘清了三种维度概念的作用：环境内在维数p_int（衡量各正交特征方向上的总变异性）、相关秩（衡量样本间的函数复杂度）以及潜在内在维数（数据中隐藏的流形结构维数）。分析表明，只要p_int ≫ log n（n为样本量），即可保证持久图能揭示潜在同调结构，且流形结构得以显现。基于此理论视角，我们重新审视Gardner等人（Nature, 2022）关于网格细胞活动存在环面结构的突破性发现：研究首次揭示该结构实际上与物理空间等距，意味着网格细胞活动能几何保真地表征真实世界。
原文摘要 · Abstract (English)
We present a generalised Hanson-Wright inequality and use it to establish new statistical insights into the geometry of data point-clouds. In the setting of a general random function model of data, we clarify the roles played by three notions of dimensionality: ambient intrinsic dimension $p_{\mathrm{int}}$, which measures total variability across orthogonal feature directions; correlation rank, which measures functional complexity across samples; and latent intrinsic dimension, which is the dimension of manifold structure hidden in data. Our analysis shows that in order for persistence diagrams to reveal latent homology and for manifold structure to emerge it is sufficient that $p_{\mathrm{int}}\gg \log n$, where $n$ is the sample size. Informed by these theoretical perspectives, we revisit the ground-breaking neuroscience discovery of toroidal structure in grid-cell activity made by Gardner et al. (Nature, 2022): our findings reveal, for the first time, evidence that this structure is in fact isometric to physical space, meaning that grid cell activity conveys a geometrically faithful representation of the real world.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。