Poisson Disk Sampling on the Grassmannnian: Applications in Subspace Optimization

To develop accurate inference algorithms on embedded manifolds such as the Grassmannian, we often employ several optimization tools and incorporate the characteristics of known manifolds as additional constraints. However, a direct analysis of the nature of functions on manifolds is rarely performed. In this paper, we propose an alternative approach to this inference by adopting a statistical pipeline that first generates an initial sampling of the manifold, and then performs subsequent analysis based on these samples. First, we introduce a better sampling technique based on dart throwing (called the Poisson disk sampling (PDS)) to effectively sample the Grassmannian. Next, using Grassmannian sparse coding, we demonstrate the improved coverage achieved by PDS. Finally, we develop a consensus approach, with Grassmann samples, to infer the optimal embeddings for linear dimensionality reduction, and show that the resulting solutions are nearly optimal.