[Paper Review] Clustering Using Isoperimetric Number of Trees

Amir Daneshgar, Ramin Javadi+1 more|arXiv (Cornell University)|Mar 19, 2012

Anomaly Detection Techniques and Applications14 references3 citations

TL;DR

This paper proposes a novel graph-based clustering algorithm that uses the isoperimetric number of minimum spanning trees to achieve exact, efficient clustering in O(n log n) time. It introduces a subpartition-based relaxation of the normalized cut problem, enabling exact k-clustering on trees and allowing simultaneous outlier detection through a residue-based outlier profile, with theoretical guarantees and strong empirical performance on benchmark datasets.

ABSTRACT

In this paper we propose a graph-based data clustering algorithm which is based on exact clustering of a minimum spanning tree in terms of a minimum isoperimetry criteria. We show that our basic clustering algorithm runs in $O(n \log n)$ and with post-processing in $O(n^2)$ (worst case) time where $n$ is the size of the data set. We also show that our generalized graph model which also allows the use of potentials at vertices can be used to extract a more detailed pack of information as the {\it outlier profile} of the data set. In this direction we show that our approach can be used to define the concept of an outlier-set in a precise way and we propose approximation algorithms for finding such sets. We also provide a comparative performance analysis of our algorithm with other related ones and we show that the new clustering algorithm (without the outlier extraction procedure) behaves quite effectively even on hard benchmarks and handmade examples.

Motivation & Objective

• To develop a theoretically grounded, efficient clustering algorithm based on graph-theoretic isoperimetry for general data clustering.
• To address the limitations of normalized cut and spectral clustering by relaxing from partitions to subpartitions, enabling natural outlier detection.
• To provide a polynomial-time exact solution for k-clustering on trees using isoperimetric criteria, overcoming NP-hardness in general graphs.
• To enable simultaneous clustering and outlier set identification through a generalized graph model with vertex potentials.
• To demonstrate superior empirical performance on hard benchmarks and artificial datasets compared to existing methods.

Proposed method

• The algorithm constructs a minimum spanning tree (MST) from the data's affinity graph to reduce computational complexity while preserving essential connectivity.
• It applies a k-isoperimetry criterion—minimizing normalized flow across subpartitions—to define optimal cluster boundaries, with a relaxation from partitions to subpartitions.
• The core method uses a greedy, bottom-up approach to identify optimal k-subpartitions on trees, leveraging the fact that the k-isoperimetry problem is solvable in polynomial time on trees.
• A residue number is introduced to quantify unclustered vertices, enabling formal definition and detection of outlier sets.
• The method incorporates local scaling and vertex potentials to enhance sensitivity to data structure and improve robustness to parameter choices.
• Post-processing in O(n²) time refines clusters and extracts the outlier profile, supporting hierarchical data similarity analysis.

Experimental results

Research questions

• RQ1 Can the k-isoperimetry problem be solved exactly and efficiently on trees, enabling a new class of exact clustering algorithms?
• RQ2 How does relaxing from partitions to subpartitions improve clustering robustness and enable natural outlier detection?
• RQ3 Can the isoperimetric approach outperform spectral clustering and normalized cut methods on hard clustering benchmarks?
• RQ4 What is the role of vertex potentials and local scaling in enhancing the algorithm’s ability to detect deviant data points?
• RQ5 How does the residue number provide a formal and computable measure for identifying outlier sets in data?

Key findings

• The k-isoperimetry problem is efficiently solvable on trees, enabling exact k-clustering in O(n log n) time, which is a key theoretical contribution.
• The algorithm achieves exact clustering on a subtree of the MST, avoiding iterative partitioning or spectral approximations used in prior methods.
• The method detects outlier sets via a residue number, with a formal reduction from 3-PARTITION proving NP-hardness of the general problem.
• Empirical results show strong performance on hard benchmarks and artificial datasets, including successful outlier extraction under varying scaling parameters.
• The algorithm maintains robustness to local scaling parameters, unlike other methods that require tuning for outlier detection.
• Theoretical analysis confirms that the cost of the optimal subpartition equals 1/(B+1) in the constructed tree, linking the solution to the 3-PARTITION decision problem.