  Modern datasets often exhibit high dimensionality, yet the data reside in
low-dimensional manifolds that can reveal underlying geometric structures
critical for data analysis. A prime example of such a dataset is a collection
of cell cycle measurements, where the inherently cyclical nature of the process
can be represented as a circle or sphere. Motivated by the need to analyze
these types of datasets, we propose a nonlinear dimension reduction method,
Spherical Rotation Component Analysis (SRCA), that incorporates geometric
information to better approximate low-dimensional manifolds. SRCA is a
versatile method designed to work in both high-dimensional and small sample
size settings. By employing spheres or ellipsoids, SRCA provides a low-rank
spherical representation of the data with general theoretic guarantees,
effectively retaining the geometric structure of the dataset during
dimensionality reduction. A comprehensive simulation study, along with a
successful application to human cell cycle data, further highlights the
advantages of SRCA compared to state-of-the-art alternatives, demonstrating its
superior performance in approximating the manifold while preserving inherent
geometric structures.
