[Paper Review] Asymptotic Behavior of Mean Partitions in Consensus Clustering

Brijnesh J. Jain|arXiv (Cornell University)|Dec 18, 2015

Bayesian Methods and Mixture Models20 references3 citations

TL;DR

This paper establishes the asymptotic consistency and asymptotic normality of mean partitions in consensus clustering by modeling partitions as points in an orbit space, leveraging Fréchet means and stochastic programming. It proves that mean partitions converge stochastically to an expected partition under mild regularity conditions, justifying finite but sufficiently large sample sizes in practice.

ABSTRACT

Although consistency is a minimum requirement of any estimator, little is known about consistency of the mean partition approach in consensus clustering. This contribution studies the asymptotic behavior of mean partitions. We show that under normal assumptions, the mean partition approach is consistent and asymptotic normal. To derive both results, we represent partitions as points of some geometric space, called orbit space. Then we draw on results from the theory of Fréchet means and stochastic programming. The asymptotic properties hold for continuous extensions of standard cluster criteria (indices). The results justify consensus clustering using finite but sufficiently large sample sizes. Furthermore, the orbit space framework provides a mathematical foundation for studying further statistical, geometrical, and analytical properties of sets of partitions.

Motivation & Objective

• To establish the asymptotic behavior of mean partitions in consensus clustering, addressing a critical gap in theoretical justification.
• To remove restrictive assumptions from prior work, such as uniqueness of the expected partition and strong concentration conditions.
• To provide a geometric foundation for analyzing sets of partitions using orbit spaces and metric structures.
• To unify the treatment of hard and soft partitions through continuous extensions of standard cluster criteria.
• To justify the use of finite but sufficiently large sample sizes in consensus clustering via consistency results.

Proposed method

• Represents partitions as points in an orbit space derived from the Euclidean space, enabling geometric and analytical treatment.
• Uses Fréchet functions to define consensus clustering as a minimization problem over the orbit space.
• Applies results from stochastic programming to derive consistency and asymptotic normality under weak regularity conditions.
• Introduces continuous extensions of standard cluster criteria (e.g., Rand index, adjusted Rand index, mutual information) to handle soft partitions.
• Employs metric structures on the orbit space to ensure continuity and convergence of the consensus function.
• Leverages the theory of Fréchet means and stochastic optimization to prove asymptotic properties without requiring compactness or strong concentration.

Experimental results

Research questions

• RQ1 Under what conditions is the mean partition in consensus clustering consistent as the sample size grows?
• RQ2 Can the asymptotic normality of mean partitions be established under general assumptions on the cluster criteria?
• RQ3 How can standard discrete cluster indices be extended continuously to support soft partitions and geometric analysis?
• RQ4 What geometric framework enables the rigorous study of statistical properties of sets of partitions?
• RQ5 Does the mean partition approach remain valid for both hard and soft partitions under the same theoretical foundation?

Key findings

• The mean partition approach is consistent under mild regularity conditions, meaning it converges stochastically to the expected partition as sample size increases.
• The mean partition is asymptotically normal, implying that its sampling distribution approaches a normal distribution for large samples.
• The orbit space framework provides a continuous geometric representation of partitions, enabling the application of advanced tools from mathematical statistics.
• Continuous extensions of standard cluster criteria (e.g., Rand index, mutual information) are constructed and shown to be continuous, supporting theoretical analysis.
• The results hold for both hard and soft partitions, unifying the theoretical treatment of consensus clustering.
• The asymptotic properties are derived using two complementary approaches: Fréchet means and stochastic programming, both yielding consistent results under different assumptions.