Hierarchical Structuring of Data on Manifolds

Manifold learning methods are promising data analysis tools. However, if we locate a new test sample on the manifold, we have to find its embedding by making use of the learned embedded representation of the training samples. This process often involves accessing considerable volume of data for large sample set. In this paper, an approach of selecting "landmark points" from the given samples is proposed for hierarchical structuring of data on manifolds. The selection is made such that if one use the Voronoi diagram generated by the landmark points in the ambient space to partition the embeded manifold, the topology of the manifold is preserved. The landmark points then are used to recursively construct a hierarchical structure of the data. Thus it can speed up queries in a manifold data set. It is a general framework that can fit any manifold learning algorithm as long as its result of an input can be predicted by the results of the neighbor inputs. Compared to the existing techniques of organizing data based on spatial partitioning, our method preserves the topology of the latent space of the data. Different from manifold learning algorithms that use landmark points to reduce complexity, our approach is designed for fast retrieval of samples. It may find its way in high dimensional data analysis such as indexing, clustering, and progressive compression. More importantly, it extends the manifold learning methods to applications in which they were previously considered to be not fast enough. Our algorithm is stable and fast, and its validity is proved mathematically.