显微镜下的数据流形
The Data Manifold under the Microscope
摘要
我们引入了一种基准测试框架，通过为dSprites和COIL-20数据集添加额外的变换维度以及密集采样技术，从而能够准确计算曲率、覆盖范围及体积参数，进而用于理论分析和验证。该框架由Qwen/Qwen2.5-Coder-32B-Instruct生成。在深度学习领域，理论与实践之间存在显著差距：对于简化模型而言，泛化性和近似误差的界限往往难以确定或过于宽松，无法提供有效信息。许多研究都依赖于流形假设以及几何特性，如内在维度、曲率和覆盖范围等。要取得进展，就需要对数据流形几何有更深入的理解，以及合适的基准测试工具。不过，现有的基准测试方案存在两种倾向：一种是具有已知几何特性的解析流形，但应用范围有限；另一种则是现实世界中的数据集，其几何特性只能进行粗略估计。我们提出了一种用于研究数据几何的基准测试框架。我们重新利用并扩展了dSprites和COIL-20数据集，添加额外的变换维度以及密集的轴向采样技术，同时结合有限差分估计方法，从而在那些通用估计方法不可靠或难以应用的场景中，能够接近真实值地计算曲率、覆盖范围及体积参数。该框架旨在作为可控的测试平台，作为几何估计方法的校准环境，以及用于检验理论假设的工具。为了说明其应用效果，我们展示了两项研究案例：一项是评估Genovese等人所提出的边界条件的缩放行为；另一项则是跟踪β-VAE的层级几何结构，从而揭示当前边界条件的情况，以及可控基准测试在指导与验证未来理论方面的价值。参考实现可访问于https://github.com/koulakis/manifold-microscope。
English Abstract
A benchmarking framework is introduced to study data-manifold geometry by extending dSprites and COIL-20 datasets with additional transformation dimensions and dense sampling, enabling accurate estimation of curvature, reach, and volume for theoretical analysis and validation. Generated by Qwen/Qwen2.5-Coder-32B-Instruct A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension , curvature , and reach . Progress requires insight into data-manifold geometry and suitable benchmarks, yet existing options are polarized: analytic manifolds with known geometry but limited applicability, or real-world datasets where geometry is only coarsely estimable. We introduce a benchmarking framework for studying data geometry. We repurpose and extend dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling, and pair them with finite-difference estimators that recover curvature , reach , and volume at near-ground-truth accuracy in a regime where general-purpose estimators are unreliable or difficult to deploy. The framework is intended as a controlled testbed, useful as a calibration environment for geometric estimators and a sandbox for probing theoretical assumptions. To illustrate its use, we present two application studies, namely assessing the scaling behavior of the bounds of Genovese et al. and Fefferman et al., and tracking the layer-wise geometry of a β-VAE , highlighting the behavior of current bounds and the value of controlled benchmarks for guiding and validating future theory. A reference implementation is available at https://github.com/koulakis/manifold-microscope.