Manifold Visualization via Short Walks

Abstract

Visualizing low-dimensional non-linear manifolds underlying high-dimensional data is a challenging data analysis problem. Different manifold visualization methods can be characterized by the associated definitions of proximity between high-dimensional data points and score functions that lead to different low-dimensional embeddings, preserving different features in the data. The geodesic distance is a popular and well-justified metric. However, it is very hard to approximate reliably from finite samples especially between far apart points. In this paper, we propose a new method called Minimap. The basic idea is to approximate local geodesic distances by shortest paths along a neighborhood graph with an additional penalizing factor based on the number of steps in the path. Embedding the resulting metric by Sammon mapping further enhances the local structures at the expense of long distances that tend to be less reliable. Experiments on real-world benchmarks suggest that Minimap can robustly visualize manifold structures.