Abstract
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.
Abstract (translated)
手写数字图像存在于高维像素空间中，但表现出强烈的几何和统计结构。本文使用三种互补的降维技术：主成分分析（PCA）、因子分析（FA）以及均匀流形近似与投影（UMAP），来研究MNIST数据集中的手写数字的潜在组织情况。不同于关注分类准确度，我们探讨每种方法如何表征固有维度、共同变化和非线性几何。 PCA揭示了主导全局方差的方向，并能利用少量组件实现高保真重构。FA将数字分解为可解释的手部书写基本元素，这些元素对应于笔画、环形和对称结构。UMAP则发现了反映不同数字类别之间平滑风格转换的非线性流形。 这些结果共同表明手写数字占据一个结构化的低维流形，并且不同的统计框架揭示了该结构的不同方面。
URL
https://arxiv.org/abs/2601.06168
