Gradient-based Hierarchical Clustering

We derive a continuous cost function for hierarchical clustering that is amenable to gradient-based optimization. Our continuous cost function is inspired by a recently proposed, discrete hierarchical cost function [5]. We present an accompanying algorithm that proceeds in two stages. In the first stage, the algorithm learns parametric routing functions at each node in a fixed hierarchy. In the second stage, the data points are clustered by routing each one from the root to a leaf. The routing function at each node may be arbitrarily complex as long as it is differentiable–e.g., a neural network. This facilitates discovery of arbitrary cluster boundaries. We present two algorithms for routing function optimization; the more efficient algorithm optimizes routers of disjoint subtrees in parallel, leading to enhanced scalability. In experiments with traditional clustering datasets, our method is competitive with other state of the art methods.