通过局部图结构回归提升流形维度估计精度
Manifold Dimension Estimation via Local Graph Structure
- 基于局部PCA坐标构建图结构回归模型
- 在合成与真实数据集上性能优于或媲美现有方法
- 适合需要高精度维度估计的科研与工程场景
现有大多数流形维度估计方法依赖于局部平坦性假设。近期，曲率修正主成分分析（CA-PCA）通过显式考虑流形曲率成为有力替代方案。受此启发，我们提出一种新框架，通过在局部PCA坐标上进行回归来捕捉流形的局部图结构。在此框架下，引入两种代表性估计器：二次嵌入（QE）和总最小二乘法（TLS）。在合成数据及真实世界数据集上的实验表明，这些方法在性能上可与当前最优方法相媲美，且常有超越表现。
原文摘要 · Abstract (English)
Most existing manifold dimension estimators rely on the assumption that the underlying manifold is locally flat within the neighborhoods under consideration. More recently, curvature-adjusted principal component analysis (CA-PCA) has emerged as a powerful alternative by explicitly accounting for the manifold's curvature. Motivated by these ideas, we propose a manifold dimension estimation framework that captures the local graph structure of the manifold through regression on local PCA coordinates. Within this framework, we introduce two representative estimators: quadratic embedding (QE) and total least squares (TLS). Experiments on both synthetic and real-world datasets demonstrate that these methods perform competitively with, and often outperform, state-of-the-art approaches.
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