Riemannian t-SNE on Several Matrix Manifolds

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 im-portant 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, building upon the established Riemannian t-SNE paradigm, we develop three manifold-specific instantiations for the Grassmann, full-rank Correlation, and fixed-rank SPSD manifolds. 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 in these settings and re-ducing distortions caused by Euclidean approximations. This work provides a systematic empirical study of geometry-aware dimensionality reduction and visualization on the three involved matrix manifolds. Extensive experimental comparisons with existing visualization methods across synthetic and benchmarking datasets demonstrate the efficacy of our pro-posal in preserving geometric properties of the data. The source code and video presentation will be released at: https://github.com/paradox-going/ManiReduce.