While classical statistics has addressed observations that are real numbers or elements of a real vector space, at present many statistical problems of high interest in the sciences address the analysis of data that consist of more complex objects, taking values in spaces that are naturally not (Euclidean) vector spaces but which still feature some geometric structure. Manifold fitting is a long-standing problem, and has finally been addressed in recent years by Fefferman et al. (2020, 2021a). We develop a method with a theory guarantee that fits a $d$-dimensional underlying manifold from noisy observations sampled in the ambient space $\mathbb{R}^D$. The new approach uses geometric structures to obtain the manifold estimator in the form of image sets via a two-step mapping approach. We prove that, under certain mild assumptions and with a sample size $N=\mathcal{O}(\sigma^{(-d+3)})$, these estimators are true $d$-dimensional smooth manifolds whose estimation error, as measured by the Hausdorff distance, is bounded by $\mathcal{O}(\sigma^2\log(1/\sigma))$ with high probability. Compared with the existing approaches proposed in Fefferman et al. (2018, 2021b); Genovese et al. (2014); Yao and Xia (2019), our method exhibits superior efficiency while attaining very low error rates with a significantly reduced sample size, which scales polynomially in $\sigma^{-1}$ and exponentially in $d$. Extensive simulations are performed to validate our theoretical results. Our findings are relevant to various fields involving high-dimensional data in statistics and machine learning. Furthermore, our method opens up new avenues for existing non-Euclidean statistical methods in the sense that it has the potential to unify them to analyze data on manifolds in the ambience space domain.

翻译：经典统计学主要处理实数或实向量空间中的观测值，而当前科学领域中许多备受关注的统计问题涉及对更复杂对象的分析，这些对象的值存在于天然非（欧几里得）向量空间但仍具有某种几何结构的空间中。流形拟合是一个长期存在的问题，近年来Fefferman等人（2020, 2021a）终于对其展开了研究。我们提出了一种具有理论保证的方法，可从环境空间$\mathbb{R}^D$中采样的带噪声观测值拟合出$d$维底层流形。该方法利用几何结构，通过两步映射方法以像集形式获得流形估计量。我们证明，在特定温和假设下，当样本量$N=\mathcal{O}(\sigma^{(-d+3)})$时，这些估计量为真正的$d$维光滑流形，其由豪斯多夫距离度量的估计误差以高概率被$\mathcal{O}(\sigma^2\log(1/\sigma))$界定量级所约束。与Fefferman等人（2018, 2021b）、Genovese等人（2014）以及Yao和Xia（2019）提出的现有方法相比，我们的方法在显著降低样本量的同时实现了极低误差率与更优效率，其中样本量关于$\sigma^{-1}$呈多项式增长，关于$d$呈指数增长。我们通过大量仿真验证了理论结果。这些发现对于统计学和机器学习中涉及高维数据的多个领域具有应用价值。此外，我们的方法为现有非欧几里得统计方法开辟了新路径，因其具有统一这些方法以分析环境空间域中流形数据的潜力。