Visualization of High-Dimensional Matrix Manifolds

Abstract: Matrix manifolds play a fundamental role in machine learning, underpinning data representations (e.g., linear subspaces and covariance matrices) and optimization procedures. These manifolds follow Riemannian geometry, where intrinsic geometric structure plays an important role in geometric learning algorithms. However, traditional visualization methods based on Euclidean assumptions often fail to respect such non-Euclidean structure, leading to distortions in the resulting embeddings. To address this limitation, we generalize the popular t-SNE paradigm to the context of Riemannian manifolds and apply it to three types of matrix manifolds, which are the Grassmann manifolds, Correlation manifolds, and Symmetric Positive Semi-Definite (SPSD) manifolds, respectively. By introducing Riemannian geodesics to define probability distributions between the original and target spaces, our method transforms high-dimensional manifold-valued data into low-dimensional embeddings, thereby respecting the intrinsic geometry of the data, with curvature-related properties implicitly reflected through geodesic distances, and reducing distortions caused by Euclidean approximations. This work provides a foundation for general-purpose dimensionality reduction of high-dimensional matrix manifolds. Extensive experimental comparisons with existing visualization methods across synthetic and benchmarking datasets demonstrate the efficacy of our proposal in preserving geometric properties of the data.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Bamdev_Mishra1

Submission Number: 6937