Spherical Rotation Dimension Reduction with Geometric Loss Functions

Modern datasets witness high-dimensionality and nontrivial geometries of spaces they live in. It would be helpful in data analysis to reduce the dimensionality while retaining the geometric structure of the dataset. Motivated by this observation, we propose a general dimension reduction method by incorporating geometric information. Our Spherical Rotation Component Analysis (SRCA) is a dimension reduction method that uses spheres or ellipsoids, to approximate a low-dimensional manifold. This method not only generalizes the Spherical Component Analysis (SPCA) method in terms of theories and algorithms and presents a comprehensive comparison of our method, as an optimization problem with theoretical guarantee and also as a structural preserving low-rank representation of data. Results relative to state-of-the-art competitors show considerable gains in ability to accurately approximate the subspace with fewer components and better structural preserving. In addition, we have pointed out that this method is a specific incarnation of a grander idea of using a geometrically induced loss function in dimension reduction tasks.