Doubly Stochastic Adaptive Neighbors Clustering via the Marcus Mapping

TL;DR

This paper introduces the Marcus Mapping, a novel method for learning sparse doubly stochastic similarity graphs for clustering, which improves computational efficiency and naturally enforces the desired number of clusters.

Contribution

The paper extends Marcus theorem to sparse matrices, proposes the ANCMM algorithm for clustering, and links Marcus mapping to optimal transport for improved efficiency.

Findings

The proposed ANCMM algorithm effectively learns doubly stochastic graphs for clustering.

Marcus mapping can be efficiently used to solve certain optimal transport problems.

Experimental results show superior performance over state-of-the-art methods.

Abstract

Clustering is a fundamental task in machine learning and data science, and similarity graph-based clustering is an important approach within this domain. Doubly stochastic symmetric similarity graphs provide numerous benefits for clustering problems and downstream tasks, yet learning such graphs remains a significant challenge. Marcus theorem states that a strictly positive symmetric matrix can be transformed into a doubly stochastic symmetric matrix by diagonal matrices. However, in clustering, learning sparse matrices is crucial for computational efficiency. We extend Marcus theorem by proposing the Marcus mapping, which indicates that certain sparse matrices can also be transformed into doubly stochastic symmetric matrices via diagonal matrices. Additionally, we introduce rank constraints into the clustering problem and propose the Doubly Stochastic Adaptive Neighbors Clustering algorithm based on the Marcus Mapping (ANCMM). This ensures that the learned graph naturally divides into the desired number of clusters. We validate the effectiveness of our algorithm through extensive comparisons with state-of-the-art algorithms. Finally, we explore the relationship between the Marcus mapping and optimal transport. We prove that the Marcus mapping solves a specific type of optimal transport problem and demonstrate that solving this problem through Marcus mapping is more efficient than directly applying optimal transport methods.