# Splitting Methods for Convex Bi-Clustering and Co-Clustering

## TL;DR

This paper introduces three efficient operator-splitting algorithms for convex bi-clustering and co-clustering, improving computational efficiency and providing theoretical analysis for large structured data clustering tasks.

## Contribution

It presents novel operator-splitting methods, including a Generalized ADMM, tailored for convex co-clustering, with demonstrated efficiency and theoretical complexity analysis.

## Key findings

- Generalized ADMM outperforms other methods on large problems
- Theoretical complexity analysis supports experimental results
- Operator-splitting methods are effective for convex co-clustering

## Abstract

Co-Clustering, the problem of simultaneously identifying clusters across multiple aspects of a data set, is a natural generalization of clustering to higher-order structured data. Recent convex formulations of bi-clustering and tensor co-clustering, which shrink estimated centroids together using a convex fusion penalty, allow for global optimality guarantees and precise theoretical analysis, but their computational properties have been less well studied. In this note, we present three efficient operator-splitting methods for the convex co-clustering problem: a standard two-block ADMM, a Generalized ADMM which avoids an expensive tensor Sylvester equation in the primal update, and a three-block ADMM based on the operator splitting scheme of Davis and Yin. Theoretical complexity analysis suggests, and experimental evidence confirms, that the Generalized ADMM is far more efficient for large problems.