Minor Surfaces are Boundaries of Mode-Based Clusters

We show that mode-based cluster boundaries exhibit themselves as minor surfaces of the data probability density function. Based on this result, we provide a connectivity measure depending on minor surface search between sample pairs. Accordingly, we build a connectivity graph among data samples. The use of graph construction is particularly demonstrated for clustering, but applications in other machine learning areas are possible. On Gaussian mixture and kernel density estimate type probability density models, we illustrate the theoretical results with examples and demonstrate that cluster boundaries between sample pairs can be detected using a line integral. We also demonstrate an example where the data distribution has a continuous line segment as its set of local maxima (not strict), for which mean-shift like gradient flow and other mode-seeking algorithms fail to identify a single cluster, while the proposed approach successfully determines this fact.