Jiaji Su

While classical data analysis 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. We develop a method with a theory guarantee that fits a d-dimensional underlying manifold from noisy observations sampled in the ambient space 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=O(σ^-(d+3)), these estimators are true d-dimensional smooth manifolds whose estimation error, as measured by the Hausdorff distance, is bounded by O(σ^2\log(1/σ)) with high probability. Compared with the existing approaches, our method exhibits superior efficiency while attaining very low error rates with a significantly reduced sample size, which scales polynomially in σ^-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 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.

Over the past decades, the increasing dimensionality of data has increased the need for effective data decomposition methods. Existing approaches, however, often rely on linear models or lack sufficient interpretability or flexibility. To address this issue, we introduce a novel nonlinear decomposition technique called the principal nested submanifolds, which builds on the foundational concepts of principal component analysis. This method exploits the local geometric information of data sets by projecting samples onto a series of nested principal submanifolds with progressively decreasing dimensions. It effectively isolates complex information within the data in a backward stepwise manner by targeting variations associated with smaller eigenvalues in local covariance matrices. Unlike previous methods, the resulting subspaces are smooth manifolds, not merely linear spaces or special shape spaces. Validated through extensive simulation studies and applied to real-world RNA sequencing data, our approach surpasses existing models in delineating intricate nonlinear structures. It provides more flexible subspace constraints that improve the extraction of significant data components and facilitate noise reduction. This innovative approach not only advances the non-Euclidean statistical analysis of data with low-dimensional intrinsic structure within Euclidean spaces, but also offers new perspectives for dealing with high-dimensional noisy data sets in fields such as bioinformatics and machine learning.