用三种方法揭示手写数字的内在结构，展现不同降维视角的互补性。
Analyzing the Structure of Handwritten Digits: A Comparative Study of PCA, Factor Analysis, and UMAP
- 通过PCA、FA和UMAP分别捕捉全局方差、笔画特征与非线性流形。
- 仅用少量主成分即可高保真重构数字图像，说明其低维结构。
- 适合对数据几何结构或降维方法比较感兴趣的读者。
手写数字图像存在于高维像素空间中，却具有显著的几何与统计结构。本文利用三种互补的降维技术——主成分分析（PCA）、因子分析（FA）和均匀流形近似与投影（UMAP），研究了MNIST数据集中手写数字的潜在组织结构。不关注分类准确率，而是探讨每种方法如何刻画内在维度、共享变异性和非线性几何。PCA揭示了主导全局方差的方向，并能用少量成分实现高保真重建；FA将数字分解为可解释的潜在书写基元，对应笔画、环状结构与对称性；UMAP揭示了反映数字类别间平滑风格过渡的非线性流形。三者共同表明：手写数字位于一个有结构的低维流形上，而不同统计框架揭示了该结构的不同侧面。
原文摘要 · Abstract (English)
Handwritten digit images lie in a high-dimensional pixel space but exhibit strong geometric and statistical structure. This paper investigates the latent organization of handwritten digits in the MNIST dataset using three complementary dimensionality reduction techniques: Principal Component Analysis (PCA), Factor Analysis (FA), and Uniform Manifold Approximation and Projection (UMAP). Rather than focusing on classification accuracy, we study how each method characterizes intrinsic dimensionality, shared variation, and nonlinear geometry. PCA reveals dominant global variance directions and enables high-fidelity reconstructions using a small number of components. FA decomposes digits into interpretable latent handwriting primitives corresponding to strokes, loops, and symmetry. UMAP uncovers nonlinear manifolds that reflect smooth stylistic transitions between digit classes. Together, these results demonstrate that handwritten digits occupy a structured low-dimensional manifold and that different statistical frameworks expose complementary aspects of this structure.
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