  Convex clustering is a promising new approach to the classical problem of
clustering, combining strong performance in empirical studies with rigorous
theoretical foundations. Despite these advantages, convex clustering has not
been widely adopted, due to its computationally intensive nature and its lack
of compelling visualizations. To address these impediments, we introduce
Algorithmic Regularization, an innovative technique for obtaining high-quality
estimates of regularization paths using an iterative one-step approximation
scheme. We justify our approach with a novel theoretical result, guaranteeing
global convergence of the approximate path to the exact solution under
easily-checked non-data-dependent assumptions. The application of algorithmic
regularization to convex clustering yields the Convex Clustering via
Algorithmic Regularization Paths (CARP) algorithm for computing the clustering
solution path. On example data sets from genomics and text analysis, CARP
delivers over a 100-fold speed-up over existing methods, while attaining a
finer approximation grid than standard methods. Furthermore, CARP enables
improved visualization of clustering solutions: the fine solution grid returned
by CARP can be used to construct a convex clustering-based dendrogram, as well
as forming the basis of a dynamic path-wise visualization based on modern web
technologies. Our methods are implemented in the open-source R package
clustRviz, available at https://github.com/DataSlingers/clustRviz.
