Academic

In the realm of classical statistics, the focus has traditionally been on handling observations that can be represented as real numbers or elements of a real vector space. However, in contemporary scientific research, there is a growing emphasis on tackling statistical problems that involve analyzing data comprising more intricate objects. These objects do not naturally conform to the properties of Euclidean vector spaces, yet they possess inherent geometric structures that warrant investigation.

Manifold fitting is a long-standing problem and has finally been addressed in recent years by  Fefferman et. al ([1,2]). In this thesis, we present a novel statistical method that offers a theoretical guarantee for fitting an underlying $d$-dimensional manifold from noisy observations sampled in the ambient space $\mathbb{R}^D$. Our approach leverages geometric structures to obtain the manifold estimator, which takes the form of image sets through a two-step mapping process. We establish that, under certain mild assumptions and with a sample size $N=\cO(\sigma^{-(d+3)})$, these estimators are true $d$-dimensional smooth manifolds whose estimation error, as measured by the Hausdorff distance, is bounded by $\cO(\sigma^2\log(1/\sigma))$ with high probability. Notably, our method outperforms existing approaches proposed in [3-6], in terms of efficiency, achieving remarkably low error rates with a considerably reduced sample size. Specifically, the required sample size scales polynomially in $\sigma^{-1}$ and exponentially in $d$. We also conduct extensive simulations to validate our theoretical findings.

In addition to the aforementioned contributions, we introduce a novel approach that harnesses the power of neural networks to fit the manifold. By leveraging the generative adversarial framework, our method learns smooth mappings between a low-dimensional latent space and the high-dimensional ambient space. The trained neural networks provide estimations for the latent manifold, enabling effective data projection onto the manifold and even generating data points residing directly within it. Through an extensive series of simulation studies and real data experiments, we thoroughly demonstrate the effectiveness and accuracy of our approach in capturing the inherent structure of the underlying manifold within the ambient space data. Notably, our method surpasses the computational efficiency limitations of previous approaches while offering control over the dimensionality and smoothness of the resulting manifold. By incorporating neural networks into the manifold fitting process, we enhance the flexibility and capability of our method, empowering it to handle complex and high-dimensional data with remarkable precision and efficiency.

The findings of our research hold significant relevance to a wide range of fields that deal with high-dimensional data in the realms of statistics and machine learning. By addressing the challenge of fitting manifolds in ambient space, our method opens up new avenues for existing non-Euclidean statistical approaches. It has the potential to unify these methods, enabling them to analyze data on manifolds within the domain of ambient spaces. This advancement not only expands the scope of statistical analysis but also provides a framework for leveraging manifold structures in high-dimensional data. By unifying non-Euclidean statistical methods, researchers and practitioners gain a comprehensive toolkit for studying and extracting valuable insights from complex datasets that exhibit manifold structures. The applicability of our method extends to a diverse range of fields, including computer vision, pattern recognition, image processing, and data mining, among others. Its potential impact lies in enhancing the understanding and utilization of manifold structures, ultimately leading to improved modeling, prediction, and decision-making in various real-world applications.