Abstract:

Manifold fitting, which offers substantial potential for efficient and accurate modeling, poses a critical challenge in non-linear data analysis. This study presents a novel approach that employs neural networks to fit the latent manifold. Leveraging the generative adversarial framework, this method learns smooth mappings between low-dimensional latent space and high-dimensional ambient space, echoing the Riemannian exponential and logarithmic maps. The well-trained neural networks provide estimations for the latent manifold, facilitate data projection onto the manifold, and even generate data points that reside directly within the manifold. Through an extensive series of simulation studies and real data experiments, we 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 exceeds the computational efficiency limitations of previous approaches and offers control over the dimensionality and smoothness of the resulting manifold. This advancement holds significant potential in the fields of statistics and computer science. The seamless integration of powerful neural network architectures with generative adversarial techniques unlocks new possibilities for manifold fitting, thereby enhancing data analysis. The implications of our findings span diverse applications, from dimensionality reduction and data visualization to generating authentic data. Collectively, our research paves the way for future advancements in non-linear data analysis and offers a beacon for subsequent scholarly pursuits. This talk is based on some results from the following references

https://www.pnas.org/doi/10.1073/pnas.2311436121 (Yao, Su and Yau, 2023),  

https://arxiv.org/abs/2304.07680 (Yao, Su, Li and Yau, 2022)  

https://arxiv.org/abs/1909.10228 (Yao and Xia, 2019).

报告人简介：

Yao Zhigang, 新加坡国立大学统计与数据科学系副教授兼终身教授。他现为哈佛大学数学科学与应用中心成员，哈佛大学统计系访问教授，清华大学YMSC访问教授，也曾作为特邀客座教授访问瑞士洛桑联邦理工大学（EPFL）等大学。 研究兴趣主要是复杂数据的统计推断。近年来专注于非欧式统计(Non-Euclidean Statistics)和低维流形学习。Yao在与Prof Shing-Tung Yau合作和帮助下，致力于推动几何和统计的交互这个全新的领域（https://cmsa.fas.harvard.edu/event/geometry-and-statistics/）。Yao也是即将在中国召开的第一届几何和统计交互的研讨会的倡导者 （https://zhigang-yao.github.io/bimsa-satellite/）。 近年来，Yao与其合作者提出在黎曼流形上重新定义传统PCA的principal flow/sub-manifold以及principal boundary等方法和理论，以及全空间下新的manifold learning方法和理论。这些方法通过考虑数据本身的非欧结构，旨在解决传统统计方法和理论中的缺陷。个人网页 https://zhigang-yao.github.io/