Largest Angle Path Distance for Multi-Manifold Clustering

We propose a novel, angle-based path metric for the multi-manifold clustering problem. This metric, which we call the largest-angle path distance (LAPD), is computed as a bottleneck path distance in a graph constructed on d-simplices of data points. When data is sampled from a collection of d-dimensional manifolds which may intersect, the method can cluster the manifolds with high accuracy and automatically detect how many manifolds are present. By leveraging fast approximation schemes for bottleneck distance, this method exhibits quasi-linear computational complexity in the number of data points. In addition to being highly scalable, the method outperforms existing algorithms in numerous numerical experiments on intersecting manifolds, and exhibits robustness with respect to noise and curvature in the data.