切空间扰动的非渐近分析
Non-Asymptotic Analysis of Tangent Space Perturbation
AI总结：
利用非渐近随机矩阵理论研究局部主成分分析估计切空间的稳定性，并给出可实际应用的扰动界限。
摘要：
为高维中靠近光滑流形的大规模含噪数据点构建高效参数化仍是一个基本问题。一种方法是通过局部切平面恢复局部参数化。主成分分析（PCA）通常是首选工具，因为它在线性子空间的无噪声样本情况下返回最优基。为了处理来自非线性流形的含噪数据样本，PCA必须在局部应用，尺度要足够小使得流形近似线性，但又足够大使得结构能从噪声中辨别。利用特征空间扰动理论和非渐近随机矩阵理论，我们研究了PCA估计的子空间随尺度的稳定性，并以高概率界定了它与真实切空间之间的夹角。通过自适应选择最小化该界限的尺度，我们的分析揭示了局部切平面恢复的合适尺度。我们还引入了几何不确定性原理，量化了噪声-曲率扰动对稳定恢复的限制。为了提供可在实践中使用的扰动界限，我们提出了插件估计，使得理论结果可以直接应用于真实数据集。
英文摘要：
Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data samples from a nonlinear manifold, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise. Using eigenspace perturbation theory and non-asymptotic random matrix theory, we study the stability of the subspace estimated by PCA as a function of scale, and bound (with high probability) the angle it forms with the true tangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals an appropriate scale for local tangent plane recovery. We also introduce a geometric uncertainty principle quantifying the limits of noise-curvature perturbation for stable recovery. With the purpose of providing perturbation bounds that can be used in practice, we propose plug-in estimates that make it possible to directly apply the theoretical results to real data sets.
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