# Dynamic Visualization and Fast Computation for Convex Clustering via Algorithmic Regularization

## TL;DR

This paper introduces Algorithmic Regularization for convex clustering, significantly improving computational speed and visualization, enabling practical and insightful clustering analysis with theoretical guarantees.

## Contribution

It proposes a novel iterative approximation scheme for convex clustering that guarantees convergence and dramatically accelerates computation while enhancing visualization capabilities.

## Key findings

- Over 100-fold speed-up over existing methods
- Finer approximation grid for clustering paths
- Enables dynamic visualization of clustering solutions

## Abstract

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.