# Spherical Rotation Dimension Reduction with Geometric Loss Functions

arXiv: https://arxiv.org/abs/2204.10975 · HTML: https://mlsanity.com/paper/2204.10975/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 79 references that are in the map.
- First posted on arXiv 23 Apr 2022.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [Efficient Manifold and Subspace Approximations with Spherelets](https://mlsanity.com/paper/1706.08263/) (2017)
- [From Geometry to Topology: Inverse Theorems for Distributed Persistence](https://mlsanity.com/paper/2101.12288/) (2021)
- [Sparse Principal Component Analysis via Variable Projection](https://mlsanity.com/paper/1804.00341/) (2018)
- [Dimensionality Reduction for Binary Data through the Projection of Natural Parameters](https://mlsanity.com/paper/1510.06112/) (2015)
- [Structure-adaptive manifold estimation](https://mlsanity.com/paper/1906.05014/) (2019)
- [Estimating densities with nonlinear support using Fisher-Gaussian kernels](https://mlsanity.com/paper/1907.05918/) (2019)
- [Generalized Penalty for Circular Coordinate Representation](https://mlsanity.com/paper/2006.02554/) (2020)
- [A Distance-preserving Matrix Sketch](https://mlsanity.com/paper/2009.03979/) (2020)

## Built on by (most cited first, 2 of 2)

- [Contrastive inverse regression for dimension reduction](https://mlsanity.com/paper/2305.12287/) (2023)
- [Efficient and Robust Bayesian Selection of Hyperparameters in Dimension Reduction for Visualization](https://mlsanity.com/paper/2306.00357/) (2023)

## Abstract (arXiv)

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.