线性化扩散映射提升高维流形数据降维效果
Linearized Diffusion Map
- 用线性近似扩散核构建新降维方法
- 在流形数据上优于PCA，尤其在高维场景
- 可直接用于非负矩阵分解，利于结构解释
我们提出线性化扩散映射（LDM），一种基于扩散映射核线性近似的新型线性降维方法。LDM结合了基于扩散的非线性方法的几何直观与PCA和经典MDS等线性嵌入的计算效率和可解释性。在合成数据（瑞士卷、超球面）和真实数据集（MNIST、COIL-20）上的大量实验表明，相较于PCA，LDM能捕捉到不同的数据几何特征。具体而言，在具有明显流形结构的数据集中，特别是在高维情形下，LDM表现优于PCA；而在以方差或噪声为主导的场景中，PCA仍更优。此外，LDM核矩阵的完全正定性使其可直接应用于非负矩阵分解（NMF），为可解释的潜在结构发现提供了可能。我们的分析表明，LDM是一种具有理论和实践潜力的新线性降维技术。
原文摘要 · Abstract (English)
We introduce the Linearized Diffusion Map (LDM), a novel linear dimensionality reduction method constructed via a linear approximation of the diffusion-map kernel. LDM integrates the geometric intuition of diffusion-based nonlinear methods with the computational simplicity, efficiency, and interpretability inherent in linear embeddings such as PCA and classical MDS. Through comprehensive experiments on synthetic datasets (Swiss roll and hyperspheres) and real-world benchmarks (MNIST and COIL-20), we illustrate that LDM captures distinct geometric features of datasets compared to PCA, offering complementary advantages. Specifically, LDM embeddings outperform PCA in datasets exhibiting explicit manifold structures, particularly in high-dimensional regimes, whereas PCA remains preferable in scenarios dominated by variance or noise. Furthermore, the complete positivity of LDM's kernel matrix allows direct applicability of Non-negative Matrix Factorization (NMF), suggesting opportunities for interpretable latent-structure discovery. Our analysis positions LDM as a valuable new linear dimensionality reduction technique with promising theoretical and practical extensions.
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